利用连分数求$q$ -切数和$q$ -割线数
The $q$-tangent and $q$-secant numbers via continued fractions
摘要：
众所周知，置换的枚举多项式的$(-1)$ -求值（分别为：（1）数的增加，使数的增加，使数的增加。正割数）。最近，Foata和Han以及Josuat-Vergès分别发现了后一结果的两个不同的$q$类似物。本文证明了几个一般的连分数展开式，从而使我们能够统一处理Josuat-Vergès的两个公式，并导出上述公式的一个新的$q$ -模拟.我们的方法是基于一个$(p,q)$ -模拟的切线和割线数通过连分数和生成函数的排列有关的五元组统计组成的定点数，弱超越数，交叉数，嵌套数和反转数。利用一种新的无序排列线性模型，给出了Josuat-Vergès公式的组合证明.
英文摘要：
It is well known that the $(-1)$-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). Recently, two distinct $q$-analogues of the latter result have been discovered by Foata and Han, and Josuat-Vergès, respectively. In this paper, we will prove some general continued fractions expansions formulae, which permits us to give a unified treatment of Josuat-Vergès' two formulae and also to derive a new $q$-analogue of the aforementioned formulae. Our approach is based on a $(p,q)$-analogue of tangent and secant numbers via continued fractions and also the generating function of permutations with respect to the quintuple statistic consisting of fixed point number, weak excedance number, crossing number, nesting number and inversion number. We also give a combinatorial proof of Josuat-Vergès' formulae by using a new linear model of derangements.
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