An Eulerian permutation statistic and generalizations

TL;DR

This paper investigates an Eulerian permutation statistic called cover, providing bijective proofs of its properties, exploring its generalizations and Mahonian partners, and offering a quasi-symmetric function proof of its joint distribution with descent and excedance statistics.

Contribution

It introduces new bijective proofs for the Eulerian nature of cover, explores its generalizations, and connects it with Mahonian statistics using advanced combinatorial methods.

Findings

Cover is proven to be Eulerian through bijective proofs

The paper establishes equal joint distributions involving cover, des, and exc

It extends the study of cover to its generalizations and Mahonian partners

Abstract

Recently, the second author studied an Eulerian statistic (called cover) in the context of convex polytopes, and proved an equal joint distribution of (cover,des) with (des,exc). In this paper, we present several direct bijective proofs that cover is Eulerian, and examine its generalizations and their Mahonian partners. We also present a quasi-symmetric function proof (suggested by Michelle Wachs) of the above equal joint distribution.