Combinatorial study on the group of parity alternating permutations

TL;DR

This paper explores the combinatorial properties of parity alternating permutations, examining their enumeration based on ascents and inversions, and revealing connections to signed Eulerian numbers.

Contribution

It provides a detailed combinatorial analysis of parity alternating permutations and uncovers their relationship with signed Eulerian numbers.

Findings

Enumeration of permutations based on ascents and inversions

Relationship between parity alternating permutations and signed Eulerian numbers

Combinatorial formulas for counting these permutations

Abstract

The combinatorial theory for the set of parity alternating permutations is expounded. In view of the numbers of ascents and inversions, several enumerative aspects of the set are investigated. In particular, it is shown that signed Eulerian numbers have intimate relationships to the set.