Asymptotics of McKay numbers for SnSn

For a partition Λ of n  , let H(Λ)H(Λ) denote its hook product. If ℓ   is prime and a⩾0a⩾0 an integer, then definepℓ(a;n)=|{Λ:|Λ|=n,a=ordℓ(H(Λ))}|. These numbers are simply related to the McKay numbers in the representation theory of the symmetric group. Using a generating function of Nakamura and the “circle method,” we determine asymptotic properties of pℓ(a;n)pℓ(a;n) and ∑aa(−1)pℓ(a;n)∑a(−1)apℓ(a;n), resolving questions of Ono. In particular we show that for fixed ℓ and n  , pℓ(a;n)pℓ(a;n) roughly fits a given distribution that is dependent on ℓ  , is centered near n−c1nlogn and has width c2n. We also give an asymptotic formula for ∑aa(−1)pℓ(a;n)∑a(−1)apℓ(a;n) that is valid whenever n is not, for any k  , within a multiplicative factor of clogℓclogℓ of ℓkℓk. This formula is of the form ±c(n)/nexp(κ(n)n) where c and κ are specific functions of n and the sign is determined by n.