Parity alternating permutations starting with an odd integer

TL;DR

This paper investigates parity alternating permutations starting with an odd number, focusing on their enumeration, subclasses like derangements, and statistical properties such as excedance.

Contribution

It introduces and analyzes parity alternating permutations beginning with an odd integer, including their counts, subclasses, and statistical properties, which is a novel focus.

Findings

Enumeration formulas for PAPs with even and odd parity.

Characterization and counting of parity alternating derangements (PADs).

Analysis of excedance statistics in PADs.

Abstract

A Parity Alternating Permutation of the set $[n] = \{1, 2,\ldots, n\}$ is a permutation with even and odd entries alternatively. We deal with parity alternating permutations having an odd entry in the first position, PAPs. We study the numbers that count the PAPs with even as well as odd parity. We also study a subclass of PAPs being derangements as well, Parity Alternating Derangements (PADs). Moreover, by considering the parity of these PADs we look into their statistical property of excedance.