From Permutations to Horizontal Visibility Patterns of Periodic Series

Periodic series of period T can be mapped into the set of permutations of [T−1]={1,2,3,…,T−1}. These permutations of period T can be classified according to the relative ordering of their elements by the horizontal visibility map. We prove that the number of horizontal visibility classes for each period T coincides with the number of triangulations of the polygon of T+1 vertices that, as is well known, is the Catalan number CT−1. We also study the robustness against Gaussian noise of the permutation patterns for each period and show that there are periodic permutations that better resist the increase of the variance of the noise.

Symmetry of Narayana Numbers and Rowvacuation of Root Posets

For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains. The W-Narayana numbers are symmetric – that is, the number of antichains of cardinality k is the same as the number of cardinality $r-k$ . However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W-Narayana numbers. Rowmotion and rowvacuation are two related operators, defined as compositions of toggles, that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev’s desired involution.

Counting pattern avoiding permutations by number of movable letters

By a movable letter within a pattern avoiding permutation, we mean one that may be transposed with its predecessor while still avoiding the pattern. In this paper, we enumerate permutations avoiding a single pattern of length three according to the number of movable letters, thereby obtaining new q- analogues of the Catalan number sequence. Indeed, we consider the joint distribution with the statistics recording the number of descents and occurrences of certain vincular patterns. To establish several of our results, we make use of the kernel method to solve the functional equations that arise.

Projections on right and left ideals of endomorphism semiring which are derivations

The aim of this paper is the investigation of the derivations in an endomorphism semiring of a finite chain. Such semiring can be represented as a simplex and its subsimplices are left ideals of the semiring. We construct projections on these left ideals and prove that they are derivations and also find the maximal subsemirings of the simplex which are the domains of the constructed derivations. Consequently, we obtain some results concerning nilpotent endomorphisms and using well-known result of Stanley we prove that order of semiring of nilpotent endomorphisms is equal to [Formula: see text], where [Formula: see text] is the [Formula: see text]th Catalan number. We consider a class of right ideals of the semiring and introduce projections on these ideals which are derivations and also find the maximal subsemirings of the simplex which are the domains of the constructed derivations. For one of these derivations [Formula: see text] and for a fixed endomorphism [Formula: see text] of a considered right ideal, the set of endomorphisms [Formula: see text] such that [Formula: see text] is denoted by [Formula: see text]. The last set is a semiring if and only if [Formula: see text] is an idempotent. The number of the semirings [Formula: see text], where [Formula: see text], is equal to [Formula: see text], which is the [Formula: see text]th Fibonacci number.

Mixed cobinary trees

We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees (MCTs). We show that the number of isomorphism classes of such trees is given by the Catalan number [Formula: see text] where [Formula: see text] is the number of internal nodes. We also consider the corresponding quiver [Formula: see text] of type [Formula: see text]. As a special case of more general known results about the relation between [Formula: see text]-vectors, representations of quivers and their semi-invariants, we explain the bijection between MCTs and the vertices of the generalized associahedron corresponding to the quiver [Formula: see text]. These results are extended to [Formula: see text]-clusters in the next paper. We give one application: a new short proof of a conjecture of Reineke using MCTs.

AN ALTERNATIVE DECOMPOSITION OF CATALAN NUMBER

A particular integer sequence derived by the convex polygon triangulation is introduced and investigated. After some underlying results are presented, the forbidden (or improper) integer values relative to the triangulation are concerned. It is understood that the forbidden sequences do not correspond to any triangulation. Some of their properties are presented. These properties are used to count the forbidden values, which is, finally, exploited in stating another decomposition of the Catalan number.

Avoidance of classical patterns by Catalan sequences

A certain subset of the words of length n over the alphabet of non-negative integers satisfying two restrictions has recently been shown to be enumerated by the Catalan number Cn-1. Members of this subset, which we will denote by W(n), have been termed Catalan words or sequences and are closely associated with the 321-avoiding permutations. Here, we consider the problem of enumerating the members of W(n) satisfying various restrictions concerning the containment of certain prescribed subsequences or patterns. Among our results, we show that the generating function counting the members Of W(n) that avoid certain patterns is always rational for four general classes of patterns. Our proofs also provide a general method of computing the generating function for all the patterns in each of the four classes. Closed form expressions in the case of three-letter patterns follow from our general results in several cases. The remaining cases for patterns of length three, which we consider in the final section, may be done by various algebraic and combinatorial methods.

Snow Leopard Permutations and Their Even and Odd Threads

Caffrey, Egge, Michel, Rubin and Ver Steegh recently introduced snow leopard permutations, which are the anti-Baxter permutations that are compatible with the doubly alternating Baxter permutations. Among other things, they showed that these permutations preserve parity, and that the number of snow leopard permutations of length $2n-1$ is the Catalan number $C_n$. In this paper we investigate the permutations that the snow leopard permutations induce on their even and odd entries; we call these the even threads and the odd threads, respectively. We give recursive bijections between these permutations and certain families of Catalan paths. We characterize the odd (resp. even) threads which form the other half of a snow leopard permutation whose even (resp. odd) thread is layered in terms of pattern avoidance, and we give a constructive bijection between the set of permutations of length $n$ which are both even threads and odd threads and the set of peakless Motzkin paths of length $n+1$. Comment: 25 pages, 6 figures. Version 3 is modified to use standard Discrete Mathematics and Theoretical Computer Science but is otherwise unchanged