  The Euler number $E_n$ (resp. Entringer number $E_{n,k}$) enumerates the
alternating (down-up) permutations of $\{1,\dots,n\}$ (resp. starting with
$k$). The Springer number $S_n$ (resp. Arnold number $S_{n,k}$) enumerates the
type $B$ alternating permutations (resp. starting with $k$). In this paper,
using bijections we first derive the counterparts in {\em Andr\'e permutations}
and {\em Simsun permutations} for the Entringer numbers $(E_{n,k})$, and then
the counterparts in {\em signed Andr\'e permutations} and {\em type $B$
increasing 1-2 trees} for the Arnold numbers $(S_{n,k})$.
