论文标题
B和D排列的签名计数以及$ t，Q $ -Euler号码
Signed Countings of Type B and D Permutations and $t,q$-Euler numbers
论文作者
论文摘要
一个经典的结果指出，所有长度$ n $的所有排列（分别乱码）的征用数量的均衡余额是Euler编号。 2010年，Josuat-Vergès提供了$ Q $ - analogue，$ Q $代表交叉点。我们将此结果扩展到B和D型的置换（分别为毁灭）。事实证明，签名计数与$ \ tan $和$ \ sec $的衍生多项式有关。 施普林格定义的施普林格数字可以被视为在每个Coxeter组上定义的Euler数字的类似物。在1992年，Arnol'd表明，经典类型A，B，D的Springer数量计算了称为Snakes的各种组合对象。在1999年，霍夫曼（Hoffman）发现，$ \ sec x $和$ \ tan x $的衍生多项式及其在某些值中评估的减法准确计算了某些类型的蛇数。然后，Josuat-Vergès研究了$（t，q）$ - 衍生多项式的类似物$ q_n（t，q）$，$ r_n（t，q）$，并表明，作为设置$ q = 1 $，多项官员是蛇的枚举者，与换人的数量有关。我们的第二个结果是找到$ q_n（t，q）$和$ r_n（t，q）$的组合解释作为蛇的枚举者，尽管结果有些混乱。
A classical result states that the parity balance of the number of excedances of all permutations (derangements, respectively) of length $n$ is the Euler number. In 2010, Josuat-Vergès gives a $q$-analogue with $q$ representing the number of crossings. We extend this result to the permutations (derangements, respectively) of type B and D. It turns out that the signed countings are related to the derivative polynomials of $\tan$ and $\sec$. Springer numbers defined by Springer can be regarded as an analogue of Euler numbers defined on every Coxeter group. In 1992 Arnol'd showed that the Springer numbers of classical types A, B, D count various combinatorial objects, called snakes. In 1999 Hoffman found that derivative polynomials of $\sec x$ and $\tan x$ and their subtraction evaluated at certain values count exactly the number of snakes of certain types. Then Josuat-Vergès studied the $(t,q)$-analogs of derivative polynomials $Q_n(t,q)$, $R_n(t,q)$ and showed that as setting $q=1$ the polynomials are enumerators of snakes with respect to the number of sign-changing. Our second result is to find combinatorial interpretations of $Q_n(t,q)$ and $R_n(t,q)$ as enumerators of the snakes, although the outcome is somewhat messy.
