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We give a bijection between permutations of length 2n and certain pairs ofDyck paths with labels on the down steps. The bijection arises from a game inwhich two players alternate selecting from a set of 2n items: the permutationencodes the players' preference ordering of the items, and the Dyck pathsencode the order in which items are selected under optimal play. We enumeratepermutations by certain statistics, AA inversions and BB inversions, which havenatural interpretations in terms of the game. We give new proofs of classicalidentities 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 arethe heights of the down steps of p.

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