  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.
