Valid plane trees: Combinatorial models for RNA secondary structures with Watson-Crick base pairs

The combinatorics of RNA plays a central role in biology. Mathematical biologists have several commonly-used models for RNA: words in a fixed alphabet (representing the primary sequence of nucleotides) and plane trees (representing the secondary structure, or folding of the RNA sequence). This paper considers an augmented version of the standard model of plane trees, one that incorporates some observed constraints on how the folding can occur. In particular we assume the alphabet consists of complementary pairs, for instance the Watson-Crick pairs A-U and C-G of RNA. Given a word in the alphabet, a valid plane tree is a tree for which, when the word is folded around the tree, each edge matches two complementary letters. Consider the graph whose vertices are valid plane trees for a fixed word and whose edges are given by Condon, Heitsch, and Hoos's local moves. We prove this graph is connected. We give an explicit algorithm to construct a valid plane tree from a primary sequence, assuming that at least one valid plane tree exists. The tree produced by our algorithm has other useful characterizations, including a uniqueness condition defined by local moves. We also study enumerative properties of valid plane trees, analyzing how the number of valid plane trees depends on the choice of sequence length and alphabet size. Finally we show that the proportion of words with at least one valid plane tree goes to zero as the word size increases. We also give some open questions.Comment: 15 pages, 10 figure

Stabilized-Interval-Free Permutations and Chord-Connected Permutations

Abstract. A stabilized-interval-free (SIF) permutation on [n], introduced by Callan, is a permutation that does not stabilize any proper interval of [n]. Such permutations are known to be the irreducibles in the decomposition of permutations along non-crossing partitions. That is, if sn denotes the number of SIF permutations on [n], S(z) = 1 + ∑ n≥1 snzn, and F (z) = 1 + ∑ n≥1 n!zn, then F (z) = S(zF (z)). This article presents, in turn, a decomposition of SIF permutations along non-crossing partitions. Specifically, by working with a convenient diagrammatic representation, given in terms of perfect matchings on alternating binary strings, we arrive at the chordconnected permutations on [n], counted by {cn}n≥1, whose generating function satisfies S(z) = C(zS(z)). The expressions at hand have immediate probabilistic interpretations, via the celebrated moment-cumulant formula of Speicher, in the context of the free probability theory of Voiculescu. The probability distributions that appear are the exponential and the complex Gaussian. Résumé. Tel que défini par Callan, une permutation sur n chiffres est dite à intervalle stabilisé si elle ne stabilise pas d’intervalle propre de [n]. Ces permutations jouent le rôle des irréductibles dans la décomposition des permutations selon les partitions non-croisées. En d’autres mots, si sn dénombre les permutations à intervalle stabilisé sur

Research on Combinatorial Statistics: Crossings and Nestings in Discrete Structures

We study the distribution of combinatorial statistics that exhibit a structure of crossings and nesting in various discrete structures, in particular, in set partitions, matchings, and fillings of moon polyominoes with entries 0 and 1. Let pi and y be two set partitions with the same number of blocks. Assume pi is a partition of [n]. For any integers l, m >̲ 0, let T (pi, l) be the set of partitions of [n + l] whose restrictions to the last n elements are isomorphic to pi, and T (pi, l, m) the subset of T (pi, l) consisting of those partitions with exactly m blocks. Similarly define T (pi, l) and T (y, l, m). We prove that if the statistic cr (ne), the number of crossings (nestings) of two edges, coincides on the sets T (pi, l) and T (pi, l) for l = 0; 1, then it coincides on T (pi, l, m) and T (y, l, m) for all l, m >̲ 0. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings. Moreover, we give a bijection between partially directed paths in the symmetric wedge y = +̲ x and matchings, which sends north steps to nestings. This gives a bijective proof of a result of E. J. Janse van Rensburg, T. Prellberg, and A. Rechnitzer that was first discovered through the corresponding generating functions: the number of partially directed paths starting at the origin confined to the symmetric wedge y = +̲ x with k north steps is equal to the number of matchings on [2n] with k nestings. Furthermore, we propose a major index statistic on 01-fillings of moon polyominoes which, when specialized to certain shapes, reduces to the major index for permutations and set partitions. We consider the set F(M, s, A) of all 01-fillings of a moon polyomino M with given column sum s whose empty rows are A, and prove that this major index has the same distribution as the number of north-east chains, which are the natural extension of inversions (resp. crossings) for permutations (resp. set partitions). Hence our result generalizes the classical equidistribution results for the permutation statistics inv and maj. Two proofs are presented. The first is an algebraic one using generating functions, and the second is a bijection on 01-fillings of moon polyominoes in the spirit of Foata's second fundamental transformation on words and permutations