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It is a classical result that the parity-balance of the number of weakexcedances of all permutations (derangements, respectively) of length $n$ isthe Euler number $E_n$, alternating in sign, if $n$ is odd (even,respectively). Josuat-Verg\`{e}s obtained a $q$-analog of the resultsrespecting the number of crossings of a permutation. One of the goals in thispaper is to extend the results to the permutations (derangements, respectively)of types B and D, on the basis of the joint distribution in statisticsexcedances, 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 alternatingpermutations of type B, called snakes. Josuat-Verg\`{e}s derived bivariatepolynomials $Q_n(t,q)$ and $R_n(t,q)$ as generalized Euler numbers viasuccessive $q$-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.

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