In this article, we compute the limit distributions of the numbers of hairpin-loops, interior-loops and bulges in k-noncrossing RNA structures. The latter are coarse-grained RNA structures allowing for cross-serial interactions, subject to the constraint that there are at most k − 1 mutually crossing arcs in the diagram representation of the molecule. We prove central limit theorems by means of studying the corresponding bivariate generating functions. These generating functions are obtained by symbolic inflation of\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\bf lv}_\textbf{\textit{k}}^{\bf 5}$$\end{document}-shapes introduced by Reidys and Wang (2009).
1. Introduction
An RNA molecule is a sequence of the four nucleotides A, G, U, C together with the Watson-Crick (A-U, G-C) and U-G base pairing rules. The sequence of bases is called the primary structure of the RNA molecule. Two bases in the primary structure that are not adjacent may form hydrogen bonds following the Watson-Crick base pairing rules. Three decades ago, Waterman and colleagues (Kleitman, 1970; Nussinov et al., 1978; Waterman, 1978) analyzed RNA secondary structures. Secondary structures are coarse-grained RNA contact structures. They can be represented as diagrams and planar graphs (Fig. 1). Diagrams are labeled graphs over the vertex set \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$[ n ] = \{ 1 , \ldots , n \}$$\end{document} with vertex degrees ≤1, represented by drawing its vertices on a horizontal line and its arcs (i, j) (i < j), in the upper half-plane (Figs. 1 and 2). Here, vertices and arcs correspond to the nucleotides A, G, U, C and Watson-Crick (A-U, G-C) and (U-G) base pairs, respectively. In a diagram, two arcs (i1, j1) and (i2, j2) are called crossing if i1 < i2 < j1 < j2 holds. Accordingly, a k-crossing is a sequence of arcs \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$(i_1 , j_1 ) , \ldots , (i_k , j_k )$$\end{document} such that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$i_1 < i_2 < \cdots < i_k < j_1 < j_2 < \cdots < j_k$$\end{document}, (Fig. 2). We call diagrams containing at most (k − 1)-crossings, k-noncrossing diagrams (k-noncrossing partial matchings). The length of an arc (i, j) is given by j − i, characterizing the minimal length of a hairpin loop. A stack of length τ is a sequence of “parallel” arcs of the form
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
((i , j ) , (i + 1 , j - 1 ) , \ldots , (i + (\tau - 1 ) , j
- (\tau - 1 ) ) ) , \tag{1.1}
\end{align*}
\end{document}
The phenylalanine tRNA secondary structure represented as 2-noncrossing diagram (top) and planar graph (bottom).
A 2-noncrossing, 2-canonical RNA structure (left) and a 3-noncrossing, 2-canonical RNA structure (right) represented as planer graphs (top) and diagrams (bottom).
and we denote it by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$S_{i , j}^{ \tau}$$\end{document}. We call an arc of length one a 1-arc. A k-noncrossing, τ-canonical RNA structure is a k-noncrossing diagram without 1-arcs, having a minimum stack-size of τ (Fig. 2). Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal T}_{k , \tau} (n )$$\end{document} denote the set of k-noncrossing, τ-canonical RNA structures of length n and let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf T}_{k , \tau} (n)$$\end{document}, which is different from the above set notation denote their number.
We next introduce the following structural elements of k-noncrossing, τ-canonical RNA structures (Figs. 3 and 4).
3-noncrossing, 6-canonical structures: the pseudoknot structure of the PrP-encoding mRNA represented as diagrams (top) and planer graphs (bottom).
The loop-types: hairpin-loop (top), interior-loop (middle), and bulge (bottom).
Let [i, j] denote an interval, i.e., a sequence of consecutive isolated vertices \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$(i , i + 1 , \ldots , j - 1 , j )$$\end{document}. We consider the following (Fig. 4):
(1) a hairpin-loop is a pair
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
( ( i , j ) , [ i + 1 , j - 1 ] ).
\end{align*}
\end{document}
(4) a stem is a sequence of stacks
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
\bigg ( S_{i_1 , j_1}^{ \tau_1} , S_{i_2 , j_2}^{ \tau_2} , \ldots
, S_{i_{s} , j_{s}}^{ \tau_{s}} \bigg )
\end{align*}
\end{document}
where the stack \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$S_{i_{m} , j_{m}}^{ \tau_m}$$\end{document} is nested in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$S_{i_{m - 1} , j_{m - 1}}^{ \tau_{m - 1}}$$\end{document}, 2 ≤ m ≤ s and there are no arcs of the form (i1 − 1,j1 + 1) and (is + τs, js − τs).
In this article, we derive the limit distributions of the numbers of hairpin-loops, interior-loops and bulges in k-noncrossing τ-canonical RNA structures,
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
\lim_ { n \to \infty } { \mathbb P } \left( \frac { { \mathbb X }
- \mu_ { k , \tau , { \mathbb X } } n } { \sqrt { n \sigma_ { k ,
\tau , { \mathbb X } } ^2 } } < x \right) = \frac { 1 } { \sqrt {
2 \pi } } \int_ { - \infty } ^ { x } \ e^ { - \frac { 1 } { 2 }
t^2 } dt , \tag {1.2}
\end{align*}
\end{document}
where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb X}$$\end{document} is a random variable counting the number of hairpin-loops, interior-loops or bulges of k-noncrossing, τ-canonical structures of length n (Fig. 5).
The distribution of hairpins (left) and bulges (right) in 3-noncrossing 1-canonical RNA structures of length n = 200. The solid curves are derived from the central limit theorem Theorem 4. The data points are obtained by uniformly generating 3-noncrossing structures (Chen et al., 2009).
2. Preliminaries
Let fk(n, ℓ) denote the number of k-noncrossing diagrams on n vertices having exactly ℓ isolated vertices. A diagram without isolated vertices is called a matching. The exponential generating function of k-noncrossing matchings satisfies the following identity (Chen et al., 2007; Grabiner and Magyar, 1993; Jin et al., 2008a)
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
{ \bf H } _k (z) = \sum_ { n \ge 0 } f_k ( 2n , 0 ) \cdot \frac
{ z^ { 2n } } { ( 2n ) ! } = \det [ I_ { i - j } ( 2z ) - I_ { i
+ j } ( 2z ) ] \mid _ { i , j = 1 } ^ { k - 1 } \tag {2.1}
\end{align*}
\end{document}
where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$I_r ( 2z ) = \sum_ { j \ge 0 } \frac { z^ { 2j + r } } { j! ( j + r ) !}$$\end{document} is the hyperbolic Bessel function of the first kind of order r. Eq. (2.1) allows us to conclude that the ordinary generating function
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
{ \bf F}_k (z) = \sum_{n \ge 0}f_k ( 2n , 0 ) z^{n}
\end{align*}
\end{document}
is D-finite (Stanley, 1980). This follows from the fact that Ir (2z) is D-finite and D-finite power series form an algebra (Stanley, 1980). Consequently, there exists some \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$e \in {\mathbb N}$$\end{document} such that
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
q_ { 0 , k } (z) \frac { d^e } { d z^e } { \bf F } _k (z) +
q_ { 1 , k } (z) \frac { d^ { e - 1 } } { d z^ { e - 1 } } {
\bf F } _k (z) + \cdots + q_ { e , k } (z) { \bf F } _k (z)
= 0 , \tag {2.2}
\end{align*}
\end{document}
where qj,k(z) are polynomials and q0,k(z) ≠ 0. The ordinary differential equations (ODE) for Fk(z), where 2 ≤ k ≤ 7 are obtained by the MAPLE package GFUN from the exact data of fk(2n, 0). They are verified by first deriving the corresponding P-recursions (Stanley, 1980) for fk(2n, 0), second transforming these P-recursions into P-recursions of fk(2n, 0)/(2n)! and third deriving the corresponding ODEs for Hk(z) and verifying that the RHS of eq. (2.1) is a solution. The key point is that any singularity of Fk(z) is contained in the set of roots of q0,k(z) (Stanley, 1980), which we denote by Rk. For 2 ≤ k ≤ 7, we give the polynomials q0,k(z) and their roots in Table 1.
Polynomialsq0,k(z) and Their Nonzero Roots Obtained by the MAPLE Package GFUN
In Jin et al. (2008b), we showed that for arbitrary k\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
f_{k}(2n,0) \, \sim \, \widetilde{c}_k \,
n^{-((k-1)^2+(k-1)/2)}\, (2(k-1))^{2n},\qquad \widetilde{c}_k>0 \tag{2.3}
\end{align*}
\end{document}
in accordance with the fact that Fk(z) has the unique dominant singularity \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\rho_k^2$$\end{document}, where ρk = 1/(2k − 2).
We next introduce a central limit theorem due to Bender (1973). It is proved by analyzing the characteristic function by the Lévy-Cramér Theorem (Theorem IX.4 in Flajolet and Sedgewick, 2009).
Theorem 1.
Suppose we are given the bivariate generating function\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
f ( z , u ) = \sum \limits_{n , t \geq 0}f ( n , t ) \ z^n \ u^t ,
\tag{2.4}
\end{align*}
\end{document}
where f (n, t) ≥0 and\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$f (n) = \sum_t f ( n , t )$$\end{document}. Let\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb X}_n$$\end{document}be a r.v. such that\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb P} ( {\mathbb X}_n = t ) = f ( n , t ) / f (n)$$\end{document}. Suppose\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
[ z^n ] f ( z , e^s ) \sim c (s) \ n^{ \alpha} \ \gamma (s) ^{ - n} \tag{2.5}
\end{align*}
\end{document}
uniformly in s in a neighborhood of 0, where c(s) is continuous and nonzero near 0, α is aconstant, and γ(s) is analytic near 0. Then there exists a pair (μ, σ) such that the normalized random variable\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
\mathbb {X}^*_{n} =\frac {{\mathbb {X}}_{n} - \mu n}
{\sqrt{n\sigma^2}} \tag{2.6}
\end{align*}
\end{document}
has asymptotically normal distribution with parameter (0, 1). That is we have\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
\lim_ { n \to \infty } { \mathbb P } \big ( { \mathbb X } ^*_ { n
} < x \big ) = \frac { 1 } { \sqrt { 2 \pi } } \int_ { - \infty }
^ { x } \ e^ { - \frac { 1 } { 2 } c^2 } dc \tag {2.7}
\end{align*}
\end{document}
The crucial points for applying Theorem 1 are (a) eq. (2.5)\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
[ z^n ] f ( z , e^s ) \sim c (s) \ n^{ \alpha} \ \gamma (s) ^{
- n} ,
\end{align*}
\end{document}
uniformly in s in a neighborhood of 0, where c(s) is continuous and nonzero near 0 and α is a constant and (b) the analyticity of γ(s) in s near 0. In the following, we have generating functions of the form Fk(ψ(z, s)). In this situation, Theorem 2 below guarantees under specific conditions
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
[ z^n ] { \bf F } _k ( \psi ( z , s ) ) \sim A (s) \ n^ { - ( (
k - 1 ) ^2 + ( k - 1 ) / 2 ) } \bigg ( \frac { 1 } { \gamma (s)
} \bigg ) ^n , \quad A (s) \ { \rm continuous } ,
\end{align*}
\end{document}
for 2 ≤ k ≤ 7. The analyticity of γ(s) is guaranteed by the analytic implicit function theorem (Flajolet and Sedgewick, 2009).
Theorem 2.
(Jin and Reidys, 2010). Suppose 2 ≤ k ≤ 7. Let ψ(z, s) be an analytic function in a domain\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
{ \cal D} = \{ ( z , s ) \mid \mid z \mid \leq r , \mid s \mid <
\epsilon \} \tag{2.9}
\end{align*}
\end{document}
such that ψ(0, s) = 0. In addition suppose γ(s) is the unique dominant singularity ofFk(ψ(z, s)) and analytic solution of\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\psi ( \gamma (s) , s ) = \rho_k^2 , \mid \gamma (s) \mid \leq r$$\end{document}, ∂zψ(γ(s), s) ≠ 0 for\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mid s \mid < \epsilon$$\end{document}. ThenFk(ψ(z, s)) has a singular expansion and\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
[z^n] {\bf F} _k (\psi (z,s)) \sim A (s) \ n^{-((k-1)^2 + ( k - 1
)/2)} \bigg(\frac{1}{\gamma (s)}\bigg)^n \quad for \ some \
continuous \ A(s) \in \mathbb{C}, \tag {2.10}
\end{align*}
\end{document}
uniformly in s contained in a small neighborhood of 0.
To keep the article self-contained, we give a direct proof of Theorem 2 in Section 5. This avoids calling upon generic results, such as the uniformity Lemma of singularity analysis (Flajolet and Sedgewick, 2009).
3. The Generating Function
In this section, we compute the bivariate generating functions of hairpin-loops, interior-loops and bulges. Let hk,τ(n, t), ik,τ(n, t) and bk,τ(n, t) denote the numbers of k-noncrossing, τ-canonical RNA structures of length n with t hairpin-loops, interior-loops and bulges. We set
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
{ \bf H}_{k , \tau} ( z , u_1 ) & = \sum_{n \geq 0} \sum_{t \geq
0} h_{k , \tau} ( n , t ) z^n \ u_1^t , \tag {3.1}
\end{align*}
\end{document}\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
{ \bf I}_{k , \tau} ( z , u_2 ) & = \sum_{n \geq 0} \sum_{t \geq
0} i_{k , \tau} ( n , t ) z^n \ u_2^t , \tag {3.2}
\end{align*}
\end{document}\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
{ \bf B}_{k , \tau} ( z , u_3 ) & = \sum_{n \geq 0} \sum_{t \geq
0}b_{k , \tau} ( n , t ) z^n \ u_3^t. \tag {3.3}
\end{align*}
\end{document}
In order to derive the above generating functions, we use symbolic enumeration (Flajolet and Sedgewick, 2009). A combinatorial class is a set of finite size with the definition of size function of its elements, whose elements are all finite size and the number of certain size elements is finite. Suppose \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal C}$$\end{document} be a combinatorial class and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$c \in { \cal C}$$\end{document}. We denote the size of c by |c|. There are two special combinatorial classes \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal E}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal Z}$$\end{document}, which contain only an element of size 0 and an element of size 1, respectively. The subset of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal C}$$\end{document} which contains all the elements of size n in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal C}$$\end{document} is denoted by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal C}_n$$\end{document}. Then the generating function of a combinatorial class \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal C}$$\end{document} is
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
{ \bf C} (z) = \sum_{c \in { \cal C}}z^{ \mid c \mid } = \sum_{n
\geq 0}C_n \ z^n , \tag{3.4}
\end{align*}
\end{document}
where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal C}_n \subset { \cal C}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$C_n = \mid { \cal C}_n \mid$$\end{document}. In particular, the generating functions of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal E}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal Z}$$\end{document} are given by E(z) = 1 and Z(z) = z. For any two combinatorial classes \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal C}$$\end{document}, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal D}$$\end{document}, we have the following operations:
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
&\bullet \ {\cal C} + { \cal D}: = { \cal C} \cup { \cal D} , \ {
\rm if} \ { \cal C} \cap { \cal D} = \emptyset \\ &\bullet \ {\cal
C} \times { \cal D}: = \{ ( c , d ) \mid c \in { \cal C}, d \in {
\cal D} \} \ { \rm and} \ { \cal C}^m: = \prod\nolimits_{i = 1}^m
{ \cal C}\\ &\bullet \ {\rm SEQ} ( { \cal C} ) = { \cal E} + {
\cal C} + { \cal C}^2 + \cdots.
\end{align*}
\end{document}
We have the following relations between the operations of combinatorial classes and the operations of their generating functions:
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
{ \cal A} &= { \cal C} + { \cal D} \rightarrow { \bf A} (z) = {
\bf C} (z) + { \bf D} (z) \tag {3.5}
\end{align*}
\end{document}\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
{ \cal A} &= { \cal C} \times { \cal D} \rightarrow { \bf A} ( z
) = { \bf C} (z) \cdot { \bf D} (z) \tag {3.6}
\end{align*}
\end{document}\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
{ \cal A} &= {\rm S\textsc{eq}} ( { \cal C} )
\rightarrow { \bf A} (z) = ( 1 - { \bf C} (z) ) ^{ - 1} , \tag
{3.7}
\end{align*}
\end{document}
where A(z), C(z), D(z) is the generating function of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal A}$$\end{document}, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal C}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal D}$$\end{document}.
Given a k-noncrossing, τ-canonical RNA structure δ, its \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\rm lv}_k^5$$\end{document}-shape, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\rm lv}_k^5 ( \delta )$$\end{document} (Reidys and Wang 2009), is obtained by first removing all isolated vertices and second collapsing any stack into a single arc (Fig. 6). By construction, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\rm lv}_k^5$$\end{document}-shapes do not preserve stack-lengths, interior loops and unpaired regions. In the following, we shall refer to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\rm lv}_k^5$$\end{document}-shape simply as shape. Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal T}_{k , \tau}$$\end{document} denote the set of k-noncrossing, τ-canonical structures and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal I}_k$$\end{document} the set of all k-noncrossing shapes and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal I}_k (m)$$\end{document} those having m 1-arcs (Fig. 6). Each stem of a k-noncrossing, τ-canonical RNA structure is mapped into an arc in its corresponding shape and all hairpin-loops are mapped into 1-arcs. Therefore, we have the surjective map,
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
\varphi: \ { \cal T}_{k , \tau} \rightarrow { \cal I}_k. \tag{3.8}
\end{align*}
\end{document}
A 3-noncrossing, 2-canonical RNA structure (top left) is mapped into its shape (top right) in two steps. A stem (blue) is mapped into a single shape-arc (blue). A hairpin-loop (red) is mapped into a shape-1-arc (red).
Indeed, for a given shape γ in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal I}_k$$\end{document}, we can derive a k-noncrossing, τ-canonical structure having arc-length ≥2, we can add arcs to each arc contained in the shape such that every resulting stack has τ arcs and insert one isolated vertex in each 1-arc. Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal I}_k ( s , m )$$\end{document} and ik(s, m) denote the set and number of the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$lv_k^5$$\end{document}-shapes of length 2s with m 1-arcs and
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
{ \bf I}_k ( x , y ) = \sum_{s \geq0} \sum_{m = 0}^{s} i_k ( s , m
) x^sy^m \tag{3.9}
\end{align*}
\end{document}
be the bivariate generating function. Furthermore, let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal I}_k (m)$$\end{document} denote the set of shapes γ having m 1-arcs. Let k, s, m be natural numbers where k ≥ 2, then the generating function Ik(x, y) (Reidys and Wang, 2009) is given by
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
{ \bf I } _k ( x , y ) = \frac { 1 + x } { 1 + 2x - xy } { \bf F
} _k \bigg ( \frac { x ( 1 + x ) } { ( 1 + 2x - xy ) ^2 } \bigg )
\tag {3.10}
\end{align*}
\end{document}
Proof. We prove the theorem via symbolic enumeration representing a k-noncrossing, τ-canonical structure as the inflation of a shape, γ. Since a structure inflated from \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\gamma \in { \cal I}_k ( s , m)$$\end{document} has exactly s stems, (2s + 1) (possibly empty) intervals of isolated vertices and m nonempty such intervals we rewrite the generating functions as
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
\begin{align*}
{ \bf H}_{k , \tau} ( z , u_1 ) & = \sum_{m \geq 0} \sum_{ \gamma
\in \ { \cal I}_k (m) }{ \bf T}_{ \gamma} ( z , u_1 , 1 , 1 ) ,\\
{ \bf I}_{k , \tau} ( z , u_2 ) & = \sum_{m \geq 0} \sum_{
\gamma \in \ { \cal I}_k (m) }{ \bf T}_{ \gamma} ( z , 1 , u_2 ,
1 ) , \\
{ \bf B}_{k , \tau} ( z , u_3 ) & = \sum_{m \geq 0}
\sum_{ \gamma \in \ { \cal I}_k (m) }{ \bf T}_{ \gamma} ( z , 1
, 1 , u_3 ) .
\end{align*}
\end{document}
where Tγ(z, u1, u2, u3) is the generating function of all k-noncrossing, τ-canonical structures with shape γ and ui(i = 1, 2, 3) are variables associated with the number of hairpin-loops, interior-loops and bulges. In order to compute the latter, we consider the inflation process: we inflate \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\gamma \in { \cal I}_k (m)$$\end{document} having s arcs, where s ≥ m, to a structure as follows:
• we inflate each arc of the shape to a stem of stacks of minimum size τ. Any isolated vertices inserted during this first inflation step separate the added stacks.
• we insert isolated vertices at the remaining (2s + 1) positions.
We inflate any shape-arc to a stack of size at least τ and subsequently add additional stacks. The latter are called induced stacks and have to be separated by means of inserting isolated vertices (Fig. 7). Note that during this first inflation step no intervals of isolated vertices, other than those necessary for separating the nested stacks are inserted. After the first inflation step we proceed inflating further by inserting only additional isolated vertices at the remaining (2s + 1) positions in which such insertions are possible. For each 1-arc at least one such isolated vertex is necessarily inserted (Fig. 8).
The first inflation step a shape (left) is inflated to a 3-noncrossing, 2-canonical structure. First, every arc in the shape is inflated to a stack of size at least two (middle), and then the shape is inflated to a new 3-noncrossing, 2-canonical structure (right) by adding one stack of size two. There are three ways to insert the isolated vertices.
The second inflation step: the structure (left) obtained in (1) in Figure 7 is inflated to a new 3-noncrossing, 2-canonical RNA structures (right) by adding isolated vertices (red).
where the indeterminants ui (i = 1, 2, 3) correspond to the labels μi, i.e., the occurrences of hairpin-loops, interior-loops and bulges. Accordingly, for any two shapes \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\gamma_1 , \gamma_2 \in { \cal I}_k (m)$$
\end{document} having s arcs, we have
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
{ \bf T}_{ \gamma_1} ( z , u_1 , u_2 , u_3 ) = { \bf T}_{
\gamma_2} ( z , u_1 , u_2 , u_3 ) . \tag{3.19}
\end{align*}
\end{document}
It now remains to observe
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
\sum_ { s \geq0 } \ \sum_ { m = 0 } ^ { s } \ i_k ( s , m ) \ x^s
\ y^m = \frac { 1 + x } { 1 + 2x - xy } { \bf F } _k \bigg (
\frac { x ( 1 + x ) } { ( 1 + 2x - xy ) ^2 } \bigg ).
\end{align*}
\end{document}
and to subsequently substitute x = η(1, 1) and y = u1z for deriving Hk,τ(z, u1). Substituting x = η(u2, 1) and y = z in we obtain Ik,τ(z, u2) and finally x = η(1, u3) and y = z produce the expression for Bk,τ(z, u3), whence the theorem. ▪
4. The Central Limit Theorem
For fixed k-noncrossing, τ-canonical structure, S, let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$${\mathbb H}_{n , k , \tau} (S) , {\mathbb I}_{n , k , \tau} (S)$$
\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$${\mathbb B}_{n , k , \tau} (S)$$
\end{document} denote the number of hairpin-loops, interior-loops, and bulges in S. Then we have the r.v.s
and\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
[ z^n ] { \bf H } _ { k , \tau } ( z , e^s ) \sim C_ { k,\tau } (
s )\ n^ { - \big ( ( k - 1 ) ^2 + \frac { k - 1 } { 2 } \big) }
\bigg ( \frac { 1 } { \gamma_ { k , \tau } (s) } \bigg ) ^n ,
\tag {4.5 }
\end{align*}
\end{document}
uniformly in s in a neighborhood of 0, where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$${C_{ k,\tau}( s)}$$
\end{document} is continuous.
Proof. The first step is to establish the existence and uniqueness of the dominant singularity γk,τ(s).
We denote
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
\vartheta_{\tau} ( z , s ) & = ( 1 - z ) ^2 ( 1 - z^2 + z^{2 \tau}
) + z^{2 \tau} - z^{2 \tau + 1}e^s , & ( 4.6 ) \\ \psi_{ \tau} ( z
, s ) & = z^{2 \tau} ( 1 - z ) ^2 ( 1 - z^2 + z^{2 \tau} )
\vartheta ( z , s ) ^{ - 2} , & ( 4.7 ) \\ \omega_{ \tau} ( z , s
) & = ( 1 - z ) ( 1 - z^2 + z^{2 \tau} ) \vartheta ( z , s ) ^{ -
1} , \tag { 4.8 }
\end{align*}
\end{document}
and consider the equations
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
\forall \ 2 \le i \le k; \qquad F_{i , \tau} ( z , s ) = \psi_{
\tau} ( z , s ) - \rho_i^2 , \tag{4.9 }
\end{align*}
\end{document}
where ρi = 1/(2i − 2). Theorem 3 and Table 1 imply that the singularities of Hk,τ(z, es) are are contained in the set of roots of
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
F_{i , \tau} ( z , s ) = 0 \quad { \rm and} \quad \vartheta_{\tau}
( z , s ) = 0 \tag{4.10 }
\end{align*}
\end{document}
where i ≤ k. Let ri,τ denote the solution of minimal modulus of
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
F_{i , \tau} ( z , 0 ) = \psi_{ \tau} ( z , 0 ) - \rho_i^2 = 0.
\tag{4.11 }
\end{align*}
\end{document}
We next verify that, for sufficiently small \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\epsilon_i > 0 , \mid z - r_{i , \tau} \mid < \epsilon_i , \mid s
\mid < \epsilon_i$$
\end{document}, the following assertions hold
• \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$ \frac { \partial } { \partial z } F_ { i , \tau } ( z , s ) \ {
\rm and } \ \frac { \partial } { \partial s } F_ { i , \tau } ( z
, s )$$
\end{document} are continuous.
The analytic implicit function theorem guarantees the existence of a unique analytic function γi,τ(s) such that, for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\mid s \mid < \epsilon_i$$
\end{document},
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
F_{i , \tau} ( \gamma_{i , \tau} (s) , s ) = 0 \quad { \rm and}
\quad \gamma_{i , \tau} ( 0 ) = r_{i , \tau}. \tag{4.12
}
\end{align*}
\end{document}
Analogously, we obtain the unique analytic function δτ(s) satisfying ϑτ(z, s) = 0 and where δτ(0) is the minimal solution of ϑτ(z, 0) = 0 for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\mid s \mid < \epsilon_{ \delta}$$
\end{document}, for some \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\epsilon_{ \delta} > 0$$
\end{document}. We next verify that the unique dominant singularity of Hk,τ(z, 1) is the minimal positive solution rk,τ of Fk,τ(z, 0) = 0 and subsequently using a continuity argument. Therefore, for sufficiently small \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\epsilon$$
\end{document} where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\epsilon < \epsilon_i$$
\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\epsilon < \epsilon_{ \delta} , \mid s \mid < \epsilon$$
\end{document}, the moduli of γi,τ(s), i < k and δτ(s) are all strictly larger than the modulus of γk,τ(s). Consequently, γk,τ(s) is the unique dominant singularity of Hk,τ(z, es) for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\mid s \mid < \epsilon$$
\end{document}.
Claim. There exists some continuous Ck,τ(s) such that, uniformly in s, for s in a neighborhood of 0
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
[z^n] {\bf H}_{k , \tau} (z , e^s) \sim C_{k,\tau} (s) \ n^{-((k -
1)^2 + \frac {k - 1} {2})} \bigg( \frac {1}{\gamma_{k , \tau} (s)}
\bigg)^n.
\end{align*}
\end{document}
To prove the Claim, let r be some positive real number such that rk,τ < r < δτ(0). For sufficiently small \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\epsilon > 0$$
\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\mid s \mid < \epsilon$$
\end{document},
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
\mid \gamma_{k , \tau} (s) \mid \leq r \quad { \rm and} \quad
\mid \delta_{\tau} (s) \mid > r.
\end{align*}
\end{document}
Then ψτ(z, s) and ωτ(z, s) are all analytic in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$${\cal D} = \{ ( z , s ) \mid \mid z \mid \leq r , \mid s \mid <
\epsilon \} $$
\end{document} and ψτ(0, s) = 0. Since γk,τ(s) is the unique dominant singularity of
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
{ \bf H}_{k , \tau} ( z , e^s ) = \omega_{ \tau} ( z , s ) \ { \bf
F}_k ( \psi_{ \tau} ( z , s ) ) ,
\end{align*}
\end{document}
satisfying
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
\psi_{ \tau} ( \gamma_{k , \tau} (s) , s ) = \rho_k^2 \quad {
\rm and} \quad \mid \gamma_{k , \tau} (s) \mid \leq r , \tag{
4.13 }
\end{align*}
\end{document}
for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\mid s \mid < \epsilon$$
\end{document}. For sufficiently small \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\epsilon > 0 ,\mid s \mid < \epsilon ,\ \frac { \partial } {
\partial z } F_ { k , \tau } ( z , s )$$
\end{document} is continuous and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$ \frac { \partial } { \partial z } F_ { k , \tau } ( r_ { k ,
\tau } , 0 ) \neq 0$$
\end{document}. Thus there exists some \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\epsilon > 0$$
\end{document}, such that for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\mid s \mid < \epsilon , \frac { \partial } { \partial z } F_ { k , \tau } ( \gamma_ { k , \tau } (s) , s ) \neq 0$$
\end{document}. According to Theorem 2, we therefore derive
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
[z^n] {\bf H}_{k , \tau} (z , e^s ) \sim C_{\ k , \tau } (s) \
n^{- ((k - 1)^2 + \frac {k - 1}{2})} \bigg ( \frac{1}{\gamma_{k ,
\tau} (s)} \bigg )^n , \tag{4.14}
\end{align*}
\end{document}
uniformly in s in a neighborhood of 0, where Ck,τ(s) is continuous.
After establishing the analogues of Proposition 1 for Ik,τ(z, u) and Bk,τ(z, u) (see at www.liebertonline.com/cmbSupplementary Material), Theorem 1 implies the following central limit theorem for the distributions of hairpin-loops, interior-loops, and bulges in k-noncrossing structures.
Theorem 4.
Let\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$k , \tau \in {\mathbb N} ,$$
\end{document} 2 ≤ k ≤ 7, 1 ≤ τ ≤ 10 and suppose the random variable\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$${\mathbb X}$$
\end{document}denotes either\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$${\mathbb H}_{n , k , \tau} , {\mathbb I}_{n , k , \tau} \ or \
{\mathbb B}_{n , k , \tau}.$$
\end{document}Then there exists a pair\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
( \mu_{k , \tau , {\mathbb X}} , \sigma_{k , \tau , {\mathbb
X}}^{2} )
\end{align*}
\end{document}
such that the normalized random variable\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$${\mathbb X}^*$$
\end{document}has asymptotically normal distribution with parameter (0, 1), where\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\mu_{k , \tau , {\mathbb X}}$$
\end{document}and\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\sigma_{k , \tau , {\mathbb X}}^2$$
\end{document}are given by\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
\mu_ { k , \tau , { \mathbb X } } = - \frac { \gamma^\prime_ { k ,
\tau , { \mathbb X } } ( 0 ) } { \gamma_ { k , \tau , { \mathbb X
} } ( 0 ) } , \qquad \sigma_ { k , \tau , { \mathbb X } } ^2 =
\bigg ( \frac { \gamma^\prime_ { k , \tau , { \mathbb X } } ( 0 )
} { \gamma_ { k , \tau , { \mathbb X } } ( 0 ) } \bigg ) ^2 -
\frac { \gamma^ { \prime \prime } _ { k , \tau , { \mathbb X } } (
0 ) } { \gamma_ { k , \tau , { \mathbb X } } ( 0 ) } , \tag {
4.15 }
\end{align*}
\end{document}
The central limit theorem for the numbers of hairpin-loops in k-noncrossing, τ-canonical structures. We list μk,τ and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\sigma_{k , \tau}^2$$
\end{document} derived from eq. (4.15).
The central limit theorem for the numbers of interior-loops in k-noncrossing, τ-canonical structures. We list μk,τ and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\sigma_{k , \tau}^2$$
\end{document} derived from eq. (4.15).
The central limit theorems for the numbers of bulges in k-noncrossing, τ-canonical structures. We list μk,τ and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\sigma_{k , \tau}^2$$
\end{document} derived from eq. (4.15).
5. Proof of Theorem 2
Proof. We consider the composite function Fk(ψ(z, s)). In view of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$[ z^n ] f ( z , s ) = \gamma^n [ z^n ] f ( \frac { z } { \gamma
} , s )$$
\end{document} it suffices to analyze the function Fk(ψ(γ(s)z, s)) and to subsequently rescale in order to obtain the correct exponential factor. For this purpose, we set
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
\widetilde{\psi} (z , s) = \psi ( \gamma (s) z , s),
\end{align*}
\end{document}
where ψ(z, s) is analytic in a domain \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$${\cal D} = \{ ( z , s ) \mid \mid z \mid \leq r , \mid s \mid <
\epsilon \} $$
\end{document}. Consequently \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\widetilde{ \psi} ( z , s )$$
\end{document} is analytic in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\mid z \mid < \widetilde{r}$$
\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\mid s \mid < \widetilde{ \epsilon}$$
\end{document}, for some \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$1 < \widetilde{r} , \ , 0 < \widetilde{ \epsilon} < \epsilon$$
\end{document}, since it's a composition of two analytic functions in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$${\cal D}$$
\end{document}. Taking its Taylor expansion at z = 1,
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
\widetilde{ \psi} ( z , s ) = \sum_{n \geq0} \widetilde{ \psi}_n (
s ) ( 1 - z ) ^n , \tag{5.1 }
\end{align*}
\end{document}
where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\widetilde{ \psi}_n (s)$$
\end{document} is analytic in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\mid s \mid < \widetilde{ \epsilon}$$
\end{document}. The singular expansion of Fk(z), 2 ≤ k ≤ 7, for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$z \rightarrow \rho_k^2$$
\end{document}, follows from the ODEs, see eq. (2.2), and is given by
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
{ \bf F}_k (z) = \begin{cases} P_k ( z - \rho_k^2 ) + c^\prime_k
( z - \rho_k^2 ) ^{ ( ( k - 1 ) ^2 + ( k - 1 ) / 2 ) - 1} \log^{}
( z - \rho_k^2 ) \big ( 1 + o ( 1 ) \big ) \\ P_k ( z - \rho_k^2 )
+ c^\prime_k ( z - \rho_k^2 ) ^{ ( ( k - 1 ) ^2 + ( k - 1 ) / 2 )
- 1} \big ( 1 + o ( 1 ) \big ) \end{cases} \tag{5.2
}
\end{align*}
\end{document}
depending on whether k is odd or even and where Pk(z) are polynomials of degree ≤ (k − 1)2 + (k − 1)/2 − 1, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$c^\prime_k$$
\end{document} constant, and ρk = 1/2(k − 1). By assumption, γ(s) is the unique analytic solution of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\psi ( \gamma (s) , s ) = \rho_k^2$$
\end{document} and by construction \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$${\bf F}_k ( \psi ( \gamma (s) z , s ) ) = { \bf F}_k (
\widetilde{ \psi} ( z , s ) )$$
\end{document}. In view of eq. (5.1), we have for z → 1 the expansion
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
\widetilde{ \psi} ( z , s ) - \rho_k^2 = \sum_{n \geq1}
\widetilde{ \psi}_n (s) ( 1 - z ) ^n = \widetilde{ \psi}_1 (s)
( 1 - z ) ( 1 + o ( 1 ) ) , \tag{5.3 }
\end{align*}
\end{document}
that is uniform in s since \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\widetilde{ \psi}_n (s)$$
\end{document} is analytic for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\mid s \mid < \widetilde{ \epsilon}$$
\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\widetilde{ \psi}_0 (s) = \psi ( \gamma (s) , s ) = \rho_k^2$$
\end{document}. As for the singular expansion of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$${\bf F}_k ( \widetilde{ \psi} ( z , s ) )$$
\end{document} we derive, substituting the eq. (5.3) into the singular expansion of Fk(z), for z → 1,
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
\begin{cases}
\widetilde{P}_k (z , s) + c_k (s) ( 1 - z ) ^{ ( ( k - 1 ) ^2 +
( k - 1 ) / 2 ) - 1} \log^{} ( 1 - z ) \big ( 1 + o ( 1 ) \big ) &
{\rm for} \ \ k \ \ {\rm odd} \\ \widetilde{P}_k ( z , s ) + c_k (
s ) ( 1 - z ) ^{ ( ( k - 1 ) ^2 + ( k - 1 ) / 2 ) - 1} \big ( 1 +
o (1) \big ) & {\rm for} \ \ k \ \ {\rm even}
\end{cases} \tag{5.4}
\end{align*}
\end{document}
where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\widetilde{P}_k ( z , s ) = P_k ( \widetilde{ \psi} ( z , s ) -
\rho_k^2 )$$
\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$c_k (s) = c^\prime_k \widetilde{ \psi}_1 (s) ^{ ( ( k - 1 )
^2 + ( k - 1 ) / 2 ) - 1}$$
\end{document} and
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
\widetilde{ \psi}_1 (s) = \partial_z \widetilde{ \psi} ( z , s )
\mid _{z = 1} = \gamma (s) \partial_z \psi ( \gamma (s) , s )
\neq 0 \quad { \rm for} \ \mid s \mid < \epsilon.
\end{align*}
\end{document}
Furthermore \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\widetilde{P}_k ( z , s )$$
\end{document} is analytic at |z| ≤ 1, whence \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$[ z^n ] \widetilde{P}_k ( z , s )$$
\end{document} is exponentially small compared to 1. Therefore, we arrive at
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
[ z^n ] { \bf F}_k ( \widetilde{ \psi} ( z , s ) ) \sim \quad
\begin{cases} [ z^n ] c_k (s) ( 1 - z ) ^{ ( ( k - 1 ) ^2 + ( k
- 1 ) / 2 ) - 1} \log^{} ( 1 - z ) \big ( 1 + o ( 1 ) \big ) \\ [
z^n ] c_k (s) ( 1 - z ) ^{ ( ( k - 1 ) ^2 + ( k - 1 ) / 2 ) - 1}
\big ( 1 + o ( 1 ) \big ) \end{cases} \tag{5.5 }
\end{align*}
\end{document}
depending on k being odd or even and uniformly in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\mid s \mid < \widetilde{ \epsilon}$$
\end{document}. We observe that ck(s) is analytic in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
$$\mid s \mid < \widetilde{ \epsilon}$$
\end{document}. Note that a dependency in the parameter s is only given in the coefficients ck(s), that are analytic in s. Standard transfer theorems (Flajolet and Sedgewick, 2009) imply that
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
[ z^n ] { \bf F}_k ( \widetilde{ \psi} ( z , s ) ) \sim A (s) \
n^{ - ( ( k - 1 ) ^2 + ( k - 1 ) / 2 ) } \quad \hbox{\rm for some}
\ A (s) \in{\mathbb C} , \tag{5.6 }
\end{align*}
\end{document}
uniformly in s contained in a small neighborhood of 0. Finally, as mentioned in the beginning of the proof, we use the scaling property of Taylor expansions in order to derive
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}
\begin{align*}
[ z^n ] { \bf F}_k ( \psi ( z , s ) ) = \bigg ( \gamma (s) \bigg
) ^{ - n} [ z^n ] { \bf F}_k ( \widetilde{ \psi} ( z , s ) ) \tag{
5.7 }
\end{align*}
\end{document}
and the proof of the Theorem is complete.
Footnotes
Acknowledgments
This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China.
Disclosure Statement
No competing financial interests exist.
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