Cyclic descents, matchings and Schur-positivity

TL;DR

This paper introduces a new descent set statistic for involutions, demonstrates its equidistribution with the standard statistic, and constructs cyclic descent extensions for various combinatorial objects, establishing Schur-positivity of related functions.

Contribution

It defines a novel descent statistic for involutions via matchings and constructs explicit cyclic descent extensions, linking geometric interpretation with algebraic properties.

Findings

New descent statistic for involutions introduced

Explicit cyclic descent extensions constructed for involutions, tableaux, and paths

Schur-positivity of associated quasisymmetric functions established

Abstract

A new descent set statistic on involutions, defined geometrically via their interpretation as matchings, is introduced in this paper, and shown to be equi-distributed with the standard one. This concept is then applied to construct explicit cyclic descent extensions on involutions, standard Young tableaux and Motzkin paths. Schur-positivity of the associated quasisymmetric functions follows.