which are given by a combinatorial process described in Section \[sec.def\].

In addition, the $p$-valuation of hook products is related to conjectures on Sylow subgroups of symmetric groups. Assume $G$ is a finite group and let $M_i(p,G)$ denote the set of ordinary irreducible characters of $G$ of degree divisible by $p^i$ but not $p^{i+1}$, for $i$ a nonnegative integer. The integers $m_i(p,G) := |M_i(p,G)|$ are called the *McKay numbers of $G$ with respect to $p$*. McKay has conjectured for instance that $m_0(p,G) =  m_0(p,N_G(H))$ if $H$ is a $p$-Sylow of $G$. This has been proved for symmetric groups by Olsson [@OlssonMcKay].

We remind the reader of the classical hook formula [@Sagan]. If $\chi^\lambda$ is the ordinary character of $\mathcal{S}_n$ (we have that $\lambda$ partitions $n$), its degree, will satisfy $
\deg \chi^\lambda = {n!}/{H(\lambda)}.
$ This quantity is also sometimes called the dimension of $\lambda$, and counts the number of standard Young tableaux of shape $\lambda$.

We thus see that $m_i(p,\mathcal{S}_n)$ is concerned with divisibility by powers of $p$ of the hook product of $\lambda$ (this is perhaps most clearly expressed in the literature in [@Kane]). Question \[q.hook\] is similarly concerned with divisibility by $p$ (for all $p$) of ratios 