Séminaire Lotharingien de Combinatoire, 78B.64 (2017), 12 pp.
Ron M. Adin, Eli Bagno and Yuval Roichman
Block Numbers of Permutations and Schur-Positivity
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.
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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