Signed Countings of Type B and D Permutations and $t,q$-Euler numbers

TL;DR

This paper extends classical parity balance results of permutation excedances to type B and D permutations, linking signed counts to derivative polynomials of tangent and secant, and explores their combinatorial interpretations.

Contribution

It introduces $q$-analogues for type B and D permutations, connecting signed counts to derivative polynomials and providing new combinatorial interpretations of these polynomials.

Findings

Signed countings relate to derivative polynomials of tangent and secant.

Extended parity balance results to type B and D permutations.

Provided combinatorial interpretations of $(t,q)$-derivatives for snakes.

Abstract

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-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 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-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 combinatorial interpretations of $Q_n(t,q)$ and $R_n(t,q)$ as enumerators of the snakes, although the outcome is somewhat messy.