Am I the last person to notice that for $latex k\geq 0$, the even moment
$latex m_{2k}=\frac{(2k)!}{2^kk!}$
of a standard gaussian random variable (with expectation zero and variance one) is the same as the index of the Weyl group of $latex \mathrm{Sp}_{2k}$ inside the Weyl group of $latex \mathrm{GL}_{2k}$ (in other words, the index of the groups of permutations of $latex 2k$ elements commuting with a fixed-point free involution among all permutations)?
If “Yes”, what else have I been missing in the same spirit?
