Partitions and McKay numbers for Sn

If Λ is a partition of an integer n, then let H (Λ) be the “hook product” of Λ. If l is prime and a≥0 is an integer, then define the partition function pl(a;n) bypl(a;n) ≔#{Λ⊢n : ordl(H(Λ))=a}.These functions occur as McKay numbers in the representation theory of the symmetric group. Using work of Nakamura, we determine several of their congruence properties. For example, we show that some of Ramanujanʹs celebrated congruences for the ordinary partition function p(n) descend to these functions.