How to decompose a permutation into a pair of labeled Dyck paths by playing a game

TL;DR

This paper introduces a bijection between permutations and labeled Dyck paths via a game, providing new combinatorial insights and proofs for classical identities involving Dyck paths and permutation statistics.

Contribution

It presents a novel bijection connecting permutations to labeled Dyck paths through a game model, and offers new proofs of classical q-identity formulas.

Findings

Established a bijection between permutations and labeled Dyck paths.

Connected permutation statistics to game outcomes and Dyck path features.

Provided new combinatorial proofs of classical q-identity formulas.

Abstract

We give a bijection between permutations of length 2n and certain pairs of Dyck paths with labels on the down steps. The bijection arises from a game in which two players alternate selecting from a set of 2n items: the permutation encodes the players' preference ordering of the items, and the Dyck paths encode the order in which items are selected under optimal play. We enumerate permutations by certain statistics, AA inversions and BB inversions, which have natural interpretations in terms of the game. We give new proofs of classical identities such as \sum_p \prod_{i=1}^n q^{h_i -1} [h_i]_q = [1]_q [3]_q ... [2n-1]_q where the sum is over all Dyck paths p of length 2n, and the h_i are the heights of the down steps of p.