Signed Countings of types B and D permutations and $t,q$Euler Numbers
Abstract
It is a classical result that the paritybalance 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). JosuatVergès 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, JosuatVergès and Kim. Springer numbers are analogous Euler numbers that count the alternating permutations of type B, called snakes. JosuatVergès 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 DOI:
 10.48550/arXiv.1708.05518
 arXiv:
 arXiv:1708.05518
 Bibcode:
 2017arXiv170805518E
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 21 pages, 3 figures. This version is revised to referee's comments. Typos corrected. To appear in Advanced in Applied Mathematics