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Actions on permutations and unimodality of descent polynomials
2008 (English)In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 29, no 2, 514-531 p.Article in journal (Refereed) Published
Abstract [en]

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 non-negative expansion in the basis {t(i) (1 + t)(n-1-2i)}(i=0)(m), m = [(n - 1)/2]. This property implies symmetry and unimodality. We prove that the action is invariant under stack sorting which strengthens recent unimodality results of Bona. We prove that the generalized permutation patterns (13-2) and (2-31) are invariant under the action and use this to prove unimodality properties for a q-analog of the Eulerian numbers recently studied by Corteel, Postnikov, Steingrimsson and Williams. We also extend the action to linear extensions of sign-graded posets to give a new proof of the unimodality of the (P, omega)-Eulerian polynomials of sign-graded posets and a combinatorial interpretations (in terms of Stembridge's peak polynomials) of the corresponding coefficients when expanded in the above basis. Finally, we prove that the statistic defined as the number of vertices of even height in the unordered decreasing tree of a permutation has the same distribution as the number of descents on any set of permutations invariant under the action. On restricting to the set of stack sortable permutations we recover a result of Kreweras.

Place, publisher, year, edition, pages
2008. Vol. 29, no 2, 514-531 p.
Keyword [en]
neggers-stanley conjecture, sortable permutations, eulerian, polynomials, charney-davis, statistics, partitions, numbers, counterexamples, enumeration, posets
URN: urn:nbn:se:kth:diva-17339DOI: 10.1016/j.ejc.2006.12.010ISI: 000253369500016OAI: diva2:335383
QC 20100525Available from: 2010-08-05 Created: 2010-08-05Bibliographically approved

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Bränden, Petter
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