A Combinatorial Proof of the Enumeration of Alternating Permutations with Given Peak Set

TL;DR

This paper provides a combinatorial proof for counting alternating permutations with a specified peak set by establishing a correspondence with cycle up-down permutations and matchings.

Contribution

It introduces a novel combinatorial approach linking cycle up-down permutations and matchings to enumerate alternating permutations with fixed peaks.

Findings

Established a bijective correspondence between permutations and matchings

Derived explicit enumeration formulas for permutations with given peak sets

Enhanced understanding of the combinatorial structure of alternating permutations

Abstract

Using the correspondence between a cycle up-down permutation and a pair of matchings, we give a combinatorial proof of the enumeration of alternating permutations according to the given peak set.