雅可比排列若干猜想的双射证明
Bijective proofs of several conjectures on Jacobi permutations
AI总结：
本文通过建立涉及多种组合结构的显式双射，证明了Henke等人提出的关于雅可比排列统计量分布的三个猜想，还得到了雅可比排列的三元指数生成函数闭式公式。
AI中文摘要：
雅可比排列由Viennot在雅可比椭圆函数的背景下提出，其计数由欧拉数决定。近期，Henke、Hoffman、Stephens、Yuan和Zhuang研究了雅可比排列的精细计数，并提出了三个关于雅可比排列上若干统计量分布的猜想。本文通过建立涉及递增偶树、递增二叉树、交替排列及安德烈排列的显式双射，证明了这些猜想。本文结果的一个亮点是雅可比排列与安德烈I型排列之间的双射，该双射将统计量对$(\text{Ascbot}, \text{last})$转换为统计量对$(\text{Desbot}, \text{first})$；其中统计量$\text{Ascbot}$（对应$\text{Desbot}$）指排列的升底（对应降底）集合，$\text{first}$（对应$\text{last}$）指排列的首字母（对应尾字母）。此外，本文研究了安德烈排列与Simsun排列上的统计量对，该统计量对与雅可比排列上的统计量对$(\text{asc}, \text{last})$等分布，其中$\text{asc}$指排列的升数。最后，本文得到了雅可比排列关于升数、比尾字母小的字母数及比尾字母大的字母数的三元指数生成函数的闭式公式。
英文摘要：
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 André permutations. One highlight of our results is a bijection between Jacobi permutations and André 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 André 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.
