Block decomposition of permutations and Schur-positivity

TL;DR

This paper explores the distribution of block numbers in 321-avoiding permutations, revealing a Schur-positivity property of their descent set generating functions, analogous to classical equi-distribution results.

Contribution

It establishes a new equi-distribution result for block numbers and last descent positions in 321-avoiding permutations, leading to Schur-positivity of related generating functions.

Findings

Distribution of left-to-right maxima matches under block number and last descent assumptions

Quasi-symmetric generating functions are Schur-positive for 321-avoiding permutations with fixed block number

Analogous to Foata-Schützenberger theorem in a new permutation class

Abstract

The block number of a permutation is the maximal number of components in its expression as a direct sum. We show that, for $321$-avoiding permutations, the set of left-to-right maxima has the same distribution when the block number is assumed to be $k$ as when the last descent of the inverse is assumed to be at position $n - k$. This result is analogous to the Foata-Sch\"utzenberger equi-distribution theorem, and implies that the quasi-symmetric generating function of descent set over $321$-avoiding permutations with a prescribed number of blocks is Schur-positive.