Petter Brändén

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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)
We study the effect on the zeros of generating functions of sequences under certain non-linear transformations. Characterizations of Pólya– Schur type are given of the transformations that preserve the property of having only real and non-positive zeros. In particular, if a polynomial a 0 + a 1 z + · · · + anz n has only real and non-positive zeros, then so(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)
The Neggers-Stanley conjecture asserts that the polynomial counting the linear extensions of a labeled finite partially ordered set by the number of descents has real zeros only. We provide counterexamples to this conjecture. A finite partially ordered set (poset) P of cardinality p is said to be labeled if its elements are identified with the integers 1,(More)