On certain McKay numbers of symmetric groups

Abstract

For primes $$\ell $$ and nonnegative integers a, we study the partition functions $$\begin{aligned} p_\ell (a;n):= \#\{\lambda \vdash n : \textrm{ord}_\ell (H(\lambda ))=a\}, \end{aligned}$$ where $$H(\lambda )$$ denotes the product of hook lengths of a partition $$\lambda $$ . These partition values arise as the McKay numbers $$m_\ell (\textrm{ord}_\ell (n!) - a; S_n)$$ in the representation theory of the symmetric group. We determine the generating functions for $$p_\ell (a;n)$$ in terms of $$p_\ell (0;n)$$ and specializations of specific D’Arcais polynomials. For $$\ell = 2$$ and 3, we give an exact formula for the $$p_\ell (a;n)$$ and prove that these values are zero for almost all n. For larger primes $$\ell $$ , the $$p_\ell (a;n)$$ are positive for sufficiently large n. Despite this positivity, we prove that $$p_\ell (a;n)$$ is almost always divisible by m for any integer m. Furthermore, with these results we prove several Ramanujan-type congruences. These include the congruences $$\begin{aligned}p_\ell (a;\ell ^k n - \delta (a,\ell )) \equiv 0 \,\,(\textrm{mod}\,\,{\ell ^{k+1}}), \end{aligned}$$ for $$0<a< \ell $$ , where $$\ell = 5, 7, 11$$ and $$\delta (a,\ell ) := (\ell ^2 - 1)/24 + a\ell $$ , which answer a question of Ono.