2020/06/13 by Liao, Hsin-Chieh
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2006.13688
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 Qn(t,q), Rn(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 Qn(t,q) and Rn(t,q) as enumerators of the snakes, although the outcome is somewhat messy.