Bijective proofs of several conjectures on Jacobi permutations

Abstract

Jacobi permutations, invented by Viennot in the context of the Jacobi elliptic functions, are counted by the Euler numbers. Recently, Henke, Hoffman, Stephens, Yuan, and Zhuang studied refined enumerations of Jacobi permutations and proposed three conjectures concerning the distribution of several statistics on Jacobi permutations. In this paper, we prove these conjectures by establishing explicit bijections involving increasing even trees, increasing binary trees, alternating permutations, and Andr\'e permutations. One highlight of our results is a bijection between Jacobi permutations and Andr\'e I permutations that transforms the pair of statistics $(\Ascbot, \last)$ to the pair of statistics $(\Desbot, \first)$. Here the statistic $\Ascbot$ (resp., $\Desbot$) denotes the set of ascent bottoms (resp., descent bottoms) of permutations, and the statistic $\first$ (resp., $\last$) denotes the first (resp., $\last$) letter of permutations. Furthermore, we investigate pairs of statistics on Andr\'e permutations and simsun permutations that are equidistributed with the pair $(\asc, \last)$ on Jacobi permutations, where $\asc$ denotes the number of ascents of permutations. Finally, we obtain a closed-form formula for the trivariate exponential generating function of Jacobi permutations with respect to the number of ascents and the numbers of letters smaller and larger than the last letter.