Séminaire Lotharingien de Combinatoire, 78B.64 (2017), 12 pp.

Ron M. Adin, Eli Bagno and Yuval Roichman

Block Numbers of Permutations and Schur-Positivity

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.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

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