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We study a group action on permutations due to Foata and Strehl and use it to prove that the descent generating polynomial of certain sets of permutations has a nonnegative expansion in the basis {t i (1 + t) n−1−2i } m i=0 , m = ⌊(n − 1)/2⌋. This property implies symmetry and unimodality. We prove that the action is invariant under stack-sorting which… (More)

We call a Stieltjes continued fraction with monic monomial numerators a Catalan continued fraction. Let e k (π) be the number of increasing subsequences of length k + 1 in the permutation π. We prove that any Cata-lan continued fraction is the multivariate generating function of a family of statistics on the 132-avoiding permutations, each consisting of a… (More)

We generalize the notion of graded posets to what we call sign-graded (labeled) posets. We prove that the W-polynomial of a sign-graded poset is symmetric and unimodal. This extends a recent result of Reiner and Welker who proved it for graded posets by associating a simplicial polytopal sphere to each graded poset. By proving that the W-polynomials of… (More)

We say that a word w on a totally ordered alphabet avoids the word v if there are no subsequences in w order-equivalent to v. In this paper we suggest a new approach to the enumeration of words on at most k letters avoiding a given pattern. By studying an automaton which for fixed k generates the words avoiding a given pattern we derive several previously… (More)

The Narayana numbers are N (n, k) = 1 n n k n k+1. There are several natural statistics on Dyck paths with a distribution given by N (n, k). We show the equidistribution of Narayana statistics by computing the flag h-vector of J(2 × n) in different ways. In the process we discover new Narayana statistics and provide co-statistics for which the Narayana… (More)

Any permutation statistic f : S → C may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: f = Σ τ λ f (τ)τ. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern p = (π, R) is an… (More)

A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large… (More)

Murota et al. have recently developed a theory of discrete convex analysis which concerns M-convex and L-convex functions on jump systems. We introduce here a family of M-concave functions arising naturally from polynomials (over the field of Puiseux series) with prescribed non-vanishing properties. This family contains several of the most studied M-concave… (More)