Signed Countings of types B and D permutations and $t,q$-Euler Numbers

TL;DR

This paper extends classical Euler number results to types B and D permutations, introducing $t,q$-analogues and providing combinatorial interpretations related to signed permutations and snakes.

Contribution

It generalizes parity-balance results to types B and D permutations and offers combinatorial interpretations of new $t,q$-Euler numbers.

Findings

Extended parity-balance results to types B and D permutations.

Derived $t,q$-analogues $Q_n(t,q)$ and $R_n(t,q)$ for signed permutations.

Provided combinatorial interpretations for these polynomials.

Abstract

It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length $n$ is the Euler number $E_n$, alternating in sign, if $n$ is odd (even, respectively). Josuat-Verg\`{e}s obtained a $q$-analog of the results respecting the number of crossings of a permutation. One of the goals in this paper is to extend the results to the permutations (derangements, respectively) of types B and D, on the basis of the joint distribution in statistics excedances, crossings and the number of negative entries obtained by Corteel, Josuat-Verg\`{e}s and Kim. Springer numbers are analogous Euler numbers that count the alternating permutations of type B, called snakes. Josuat-Verg\`{e}s derived bivariate polynomials $Q_n(t,q)$ and $R_n(t,q)$ as generalized Euler numbers via successive $q$-derivatives and multiplications by $t$ on polynomials in $t$. The other goal in this paper is to give a combinatorial interpretation of $Q_n(t,q)$ and $R_n(t,q)$ as the enumerators of the snakes with restrictions.