Convex Polytopes and Enumeration

AbstractThis is an expository paper on connections between enumerative combinatorics and convex polytopes. It aims to give an essentially self-contained overview of five specific instances when enumerative combinatorics and convex polytopes arise jointly in problems whose initial formulation lies in only one of these two subjects. These examples constitute only a sample of such instances occurring in the work of several authors. On the enumerative side, they involved ordered graphical sequences, combinatorial statistics on the symmetric and hyperoctahedral groups, lattice paths, Baxter, André, and simsun permutations,q-Catalan andq-Schröder numbers. From the subject of polytopes, the examples involve the Ehrhart polynomial, the permutohedron, the associahedron, polytopes arising as intersections of cubes and simplices with half-spaces, and thecd-index of a polytope

On the joint distributions of succession and Eulerian statistics

The motivation of this paper is to investigate the joint distribution of succession and Eulerian statistics. We first investigate the enumerators for the joint distribution of descents, big ascents and successions over all permutations in the symmetric group. As an generalization a result of Diaconis-Evans-Graham (Adv. in Appl. Math., 61 (2014), 102-124), we show that two triple set-valued statistics of permutations are equidistributed on symmetric groups. We then introduce the definition of proper left-to-right minimum, and discover that the joint distribution of the succession and proper left-to-right minimum statistics over permutations is a symmetric distribution. In the final part, we discuss the relationship between the fix and cyc (p,q)-Eulerian polynomials and the joint distribution of succession and Eulerian-type statistics. In particular, we give a concise derivation of the generating function for a six-variable Eulerian polynomials.Comment: 21 pages. arXiv admin note: text overlap with arXiv:2002.0693

Permutations Avoiding a Simsun Pattern

A permutation \u3c0 avoids the simsun pattern \u3c4 if \u3c0 avoids the consecutive pattern \u3c4 and the same condition applies to the restriction of \u3c0 to any interval [k]. Permutations avoiding the simsun pattern 321 are the usual simsun permutation introduced by Simion and Sundaram. Deutsch and Elizalde enumerated the set of simsun permutations that avoid in addition any set of patterns of length 3 in the classical sense. In this paper we enumerate the set of permutations avoiding any other simsun pattern of length 3 together with any set of classical patterns of length 3. The main tool in the proofs is a massive use of a bijection between permutations and increasing binary trees

Three New Refined Arnold Families

The Springer numbers, introduced by Arnold, are generalizations of Euler numbers in the sense of Coxeter groups. They appear as the row sums of a double triangular array $(v_{n,k})$ of integers, $1\leq|k|\leq n$, defined recursively by a boustrophedon algorithm. We say a sequence of combinatorial objects $(X_{n,k})$ is an Arnold family if $X_{n,k}$ is counted by $v_{n,k}$. A polynomial refinement $V_{n,k}(t)$ of $v_{n,k}$, together with the combinatorial interpretations in several combinatorial structures was introduced by Eu and Fu recently. In this paper, we provide three new Arnold families of combinatorial objects, namely the cycle-up-down permutations, the valley signed permutations and Knuth's flip equivalences on permutations. We shall find corresponding statistics to realize the refined polynomial arrays

Polytopes, generating functions, and new statistics related to descents and inversions in permutations

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.Includes bibliographical references (p. 75-76).We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation [sigma] = [sigma] 1 [sigma] 2 an defined as the set of indices i such that either i is odd and ai > ui+l, or i is even and au < au+l. We show that this statistic is equidistributed with the 3-descent set statistic on permutations [sigma] = [sigma] 1 [sigma] 2 ... [sigma] n+1 with al = 1, defined to be the set of indices i such that the triple [sigma] i [sigma] i + [sigma] i +2 forms an odd permutation of size 3. We then introduce Mahonian inversion statistics corresponding to the two new variations of descents and show that the joint distributions of the resulting descent-inversion pairs are the same. We examine the generating functions involving alternating Eulerian polynomials, defined by analogy with the classical Eulerian polynomials ... using alternating descents. By looking at the number of alternating inversions in alternating (down-up) permutations, we obtain a new qanalog of the Euler number En and show how it emerges in a q-analog of an identity expressing E, as a weighted sum of Dyck paths. Other parts of this thesis are devoted to polytopes relevant to the descent statistic. One such polytope is a "signed" version of the Pitman-Stanley parking function polytope, which can be viewed as a generalization of the chain polytope of the zigzag poset. We also discuss the family of descent polytopes, also known as order polytopes of ribbon posets, giving ways to compute their f-vectors and looking further into their combinatorial structure.by Denis Chebikin.Ph.D