- 學位論文
Signed Countings of Type B and D Permutations and t,q-Euler numbers
Signed Countings of Type B and D Permutations and t,q-Euler numbers
摘要
並列摘要
A classical result states that the parity balance of number of excedances of all permutations (derangements, respectivly) of length $n$ is the Euler number. In 2010, Josuat-Verg`{e}s 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 counting are related to the derivative polynomials of $ an$ 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 $ an x$ and their subtraction evaluated at certain values count exactly the number of snakes of certain types. Then Josuat-Verg`{e}s 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 a combinatorial interpretations of $Q_n(t,q)$ and $R_n(t,q)$ as enumerator of the snakes, although the outcome is somewhat messy.