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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 characterize all linear operators on finite or infinite-dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region Â ރ for arbitrary closed circular domains (i.e., images of the closed unit disk under a Möbius transformation) and their boundaries. This provides a natural framework for dealing with several… (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)

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)

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)

- Petter Brändén
- 2010

A hyperbolic program is an optimization problem of the form minimize c T x such that Ax = b and x ∈ Λ + , where c ∈ R n , Ax = b is a system of linear equations and Λ + is the closure of a so called hyperbolicity cone. Hyperbolic programming generalizes semidefinite programming, but it is not known to what extent since it is not known how general the… (More)