Block numbers of permutations and Schur-positivity

Ron M. Adin; Eli Bagno; Yuval Roichman

Block numbers of permutations and Schur-positivity

Ron M. Adin, Eli Bagno, Yuval Roichman

Department of Mathematics

Jerusalem College of Technology

Research output: Contribution to conference › Paper › peer-review

Abstract

The block number of a permutation is the maximal number of components in its expression as a direct sum. We show that the distribution of the set of left-to-right-maxima over 321-avoiding permutations with a given block number k is equal to the distribution of this set over 321-avoiding permutations with the last descent of the inverse permutation at position n - k. This result is analogous to the Foata-Schützenberger equi-distribution theorem, and implies Schur-positivity of the quasi-symmetric generating function of descent set over 321-avoiding permutations with a prescribed block number.

Original language	English
State	Published - 2006
Event	29th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2017 - London, United Kingdom Duration: 9 Jul 2017 → 13 Jul 2017

Conference

Conference	29th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2017
Country/Territory	United Kingdom
City	London
Period	9/07/17 → 13/07/17

Bibliographical note

Publisher Copyright:
© 29th international conference on Formal Power Series and Algebraic Combinatorics. All rights reserved.

Keywords

Pattern avoidance
Permutation statistics
Quasi-symmetric function
Schur positivity

Cite this

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AU - Bagno, Eli

AU - Roichman, Yuval

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AB - The block number of a permutation is the maximal number of components in its expression as a direct sum. We show that the distribution of the set of left-to-right-maxima over 321-avoiding permutations with a given block number k is equal to the distribution of this set over 321-avoiding permutations with the last descent of the inverse permutation at position n - k. This result is analogous to the Foata-Schützenberger equi-distribution theorem, and implies Schur-positivity of the quasi-symmetric generating function of descent set over 321-avoiding permutations with a prescribed block number.

KW - Pattern avoidance

KW - Permutation statistics

KW - Quasi-symmetric function

KW - Schur positivity

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T2 - 29th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2017

Y2 - 9 July 2017 through 13 July 2017

ER -

Block numbers of permutations and Schur-positivity

Abstract

Conference

Bibliographical note

Keywords

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Cite this