On cyclic Schur-positive sets of permutation

TL;DR

This paper introduces cyclic Schur-positivity for permutation sets, extending classical Schur-positivity, and demonstrates that certain rotation-invariant classes are cyclic Schur-positive, revealing new combinatorial phenomena and solutions.

Contribution

It defines cyclic Schur-positivity, proves that rotation-invariant permutation classes are cyclic Schur-positive, and uncovers new combinatorial equidistribution phenomena.

Findings

Certain rotation-invariant permutation classes are cyclic Schur-positive.

The proof reveals a new equidistribution phenomenon of descent sets.

Provides affirmative solutions to existing conjectures and new examples of Schur-positive sets.

Abstract

We introduce a notion of {\em cyclic Schur-positivity} for sets of permutations, which naturally extends the classical notion of Schur-positivity, and it involves the existence of a bijection from permutations to standard Young tableaux that preserves the cyclic descent set. Cyclic Schur-positive sets of permutations are always Schur-positive, but the converse does not hold, as exemplified by inverse descent classes, Knuth classes and conjugacy classes. In this paper we show that certain classes of permutations invariant under either horizontal or vertical rotation are cyclic Schur-positive. The proof unveils a new equidistribution phenomenon of descent sets on permutations, provides affirmative solutions to conjectures from [9] and [2], and yields new examples of Schur-positive sets.