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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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