Results 1  10
of
52
Combinatorics of RNA structures with pseudoknots. Bull.Math.Biol
 Bull.Math.Biol
, 2007
"... Abstract. In this paper we derive the generating function of RNA structures with pseudoknots. We enumerate all knoncrossing RNA pseudoknot structures categorized by their maximal sets of mutually intersecting arcs. In addition we enumerate pseudoknot structures over circular RNA. For 3noncrossing ..."
Abstract

Cited by 20 (15 self)
 Add to MetaCart
Abstract. In this paper we derive the generating function of RNA structures with pseudoknots. We enumerate all knoncrossing RNA pseudoknot structures categorized by their maximal sets of mutually intersecting arcs. In addition we enumerate pseudoknot structures over circular RNA. For 3noncrossing RNA structures and RNA secondary structures we present a novel 4term recursion formula and a 2term recursion, respectively. Furthermore we enumerate for arbitrary k all knoncrossing, restricted RNA structures i.e. knoncrossing RNA structures without 2arcs i.e. arcs of the form (i, i + 2), for 1 ≤ i ≤ n − 2. 1.
Linked Partitions and Linked Cycles
"... The notion of noncrossing linked partition arose from the study of certain transforms in free probability theory. It is known that the number of noncrossing linked partitions of [n+1] is equal to the nth large Schröder number rn, which counts the number of Schröder paths. In this paper we give a bi ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
The notion of noncrossing linked partition arose from the study of certain transforms in free probability theory. It is known that the number of noncrossing linked partitions of [n+1] is equal to the nth large Schröder number rn, which counts the number of Schröder paths. In this paper we give a bijective proof of this result. Then we introduce the structures of linked partitions and linked cycles. We present various combinatorial properties of noncrossing linked partitions, linked partitions, and linked cycles, and connect them to other combinatorial structures and results, including increasing trees, partial matchings, kStirling numbers of the second kind, and the symmetry between crossings and nestings over certain linear graphs.
A bijection between 2triangulations and pairs of noncrossing Dyck paths
 Journal of Combinatorial Theory Series A
, 2006
"... A ktriangulation of a convex polygon is a maximal set of diagonals so that no k + 1 of them mutually cross in their interiors. We present a bijection between 2triangulations of a convex ngon and pairs of noncrossing Dyck paths of length 2(n − 4). This solves the problem of finding a bijective pr ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
A ktriangulation of a convex polygon is a maximal set of diagonals so that no k + 1 of them mutually cross in their interiors. We present a bijection between 2triangulations of a convex ngon and pairs of noncrossing Dyck paths of length 2(n − 4). This solves the problem of finding a bijective proof of a result of Jonsson for the case k = 2. We obtain the bijection by constructing isomorphic generating trees for the sets of 2triangulations and pairs of noncrossing Dyck paths. 1.
Crossings and Nestings of Two Edges in Set Partitions
"... AMS subject classifications. 05A18, 05A15 Let π and λ be two set partitions with the same number of blocks. Assume π is a partition of [n]. For any integer l, m ≥ 0, let T (π, l) be the set of partitions of [n + l] whose restrictions to the last n elements are isomorphic to π, and T (π, l, m) the su ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
AMS subject classifications. 05A18, 05A15 Let π and λ be two set partitions with the same number of blocks. Assume π is a partition of [n]. For any integer l, m ≥ 0, let T (π, l) be the set of partitions of [n + l] whose restrictions to the last n elements are isomorphic to π, and T (π, l, m) the subset of T (π, l) consisting of those partitions with exactly m blocks. Similarly define T (λ, l) and T (λ, l, m). We prove that if the statistic cr (ne), the number of crossings (nestings) of two edges, coincides on the sets T (π, l) and T (λ, l) for l = 0, 1, then it coincides on T (π, l, m) and T (λ, l, m) for all l, m ≥ 0. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings. 1 Introduction and Statement of Main Result In a recent paper [5], Klazar studied distributions of the numbers of crossings and nestings of two edges in (perfect) matchings. All matchings form an infinite tree T rooted at the empty matching ∅, in which the children of a matching M are the matchings obtained from M by adding to M in all possible ways a new first edge. Given two matchings M and N on [2n], Klazar decided when
Reidys, Asymptotic enumeration of RNA structures with pseudoknots
, 2008
"... Abstract. In this paper we present the asymptotic enumeration of RNA structures with pseudoknots. We develop a general framework for the computation of exponential growth rate and the sub exponential factors for knoncrossing RNA structures. Our results are based on the generating function for the n ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Abstract. In this paper we present the asymptotic enumeration of RNA structures with pseudoknots. We develop a general framework for the computation of exponential growth rate and the sub exponential factors for knoncrossing RNA structures. Our results are based on the generating function for the number of knoncrossing RNA pseudoknot structures, Sk(n), derived in [17], where k − 1 denotes the maximal size of sets of mutually intersecting bonds. We prove a functional equation for the generating function P n≥0 Sk(n)z n and obtain for k = 2 and k = 3 the analytic continuation and singular expansions, respectively. It is implicit in our results that for arbitrary k singular expansions exist and via transfer theorems of analytic combinatorics we obtain asymptotic expression for the coefficients. We explicitly derive the asymptotic expressions for 2 and 3noncrossing RNA structures. Our main result is the derivation of the formula
Pseudoknot RNA structures with arclenght � 4
 J. Comp. bio
, 2008
"... Abstract. In this paper we study knoncrossing RNA structures with arclength ≥ 3, i.e. RNA molecules in which for any i, the nucleotides labeled i and i + j (j = 1,2) cannot form a bond and in which there are at most k − 1 mutually crossing arcs. Let Sk,3(n) denote their number. Based on a novel fu ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
Abstract. In this paper we study knoncrossing RNA structures with arclength ≥ 3, i.e. RNA molecules in which for any i, the nucleotides labeled i and i + j (j = 1,2) cannot form a bond and in which there are at most k − 1 mutually crossing arcs. Let Sk,3(n) denote their number. Based on a novel functional equation for the generating function P n≥0 Sk,3(n)z n, we derive for arbitrary k ≥ 3 exponential growth factors and for k = 3 the subexponential factor. Our main result is the derivation of the formula S3,3(n) ∼ 6.11170·4! n(n−1)...(n−4) 4.54920n. 1.
EFFICIENT COUNTING AND ASYMPTOTICS OF kNONCROSSING TANGLEDDIAGRAMS
, 2008
"... Abstract. In this paper we enumerate knoncrossing tangleddiagrams. A tangleddiagram is a labeled graph whose vertices are 1,..., n have degree ≤ 2, and are arranged in increasing order in a horizontal line. Its arcs are drawn in the upper halfplane with a particular notion of crossings and nestin ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Abstract. In this paper we enumerate knoncrossing tangleddiagrams. A tangleddiagram is a labeled graph whose vertices are 1,..., n have degree ≤ 2, and are arranged in increasing order in a horizontal line. Its arcs are drawn in the upper halfplane with a particular notion of crossings and nestings. Our main result is the asymptotic formula for the number of knoncrossing tangleddiagrams Tk(n) ∼ ck n −((k−1)2 +(k−1)/2) (4(k − 1) 2 + 2(k − 1) + 1) n for some ck> 0. 1. Tangled diagrams as molecules or walks In this paper we show how to compute the numbers of knoncrossing tangleddiagrams and prove the asymptotic formula (1.1) Tk(n) ∼ ck n −((k−1)2 +(k−1)/2) (4(k − 1) 2 + 2(k − 1) + 1) n, ck> 0. This article is accompanied by a Maple package TANGLE, downloadable from the webpage
Folding 3noncrossing RNA pseudoknot structures
 J. Comput. Biol
"... Abstract. In this paper we present a selfcontained analysis and description of the novel ab initio folding algorithm cross, which generates the minimum free energy (mfe), 3noncrossing, σcanonical RNA structure. Here an RNA structure is 3noncrossing if it does not contain more than three mutually ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Abstract. In this paper we present a selfcontained analysis and description of the novel ab initio folding algorithm cross, which generates the minimum free energy (mfe), 3noncrossing, σcanonical RNA structure. Here an RNA structure is 3noncrossing if it does not contain more than three mutually crossing arcs and σcanonical, if each of its stacks has size greater or equal than σ. Our notion of mfestructure is based on a specific concept of pseudoknots and respective loopbased energy parameters. The algorithm decomposes into three parts: the first is the inductive construction of motifs and shadows, the second is the generation of the skeletatrees rooted in irreducible shadows and the third is the saturation of skeleta via context dependent dynamic programming routines. 1. Introduction and
On kcrossings and knestings of permutations
 Proc. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS proc
, 2010
"... Abstract. We introduce kcrossings and knestings of permutations. We show that the crossing number and the nesting number of permutations have a symmetric joint distribution. As a corollary, the number of knoncrossing permutations is equal to the number of knonnesting permutations. We also provid ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Abstract. We introduce kcrossings and knestings of permutations. We show that the crossing number and the nesting number of permutations have a symmetric joint distribution. As a corollary, the number of knoncrossing permutations is equal to the number of knonnesting permutations. We also provide some enumerative results for knoncrossing permutations for some values of k. Résumé. Nous introduisons les kchevauchement d’arcs et les kempilements d’arcs de permutations. Nous montrons que l’index de chevauchement et l’index de empilement ont une distribution conjointe symétrique pour les permutations de taille n. Comme corollaire, nous obtenons que le nombre de permutations n’ayant pas un kchevauchement est égal au nombre de permutations n’ayant un kempilement. Nous fournissons également quelques résultats énumératifs.