Asymptotics of McKay numbers for Sn Original Research Article

For a partition Λ of n, let H(Λ) denote its hook product. If ℓ is prime and agreater-or-equal, slanted0 an integer, then defineimage 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) and ∑a(−1)apℓ(a;n), resolving questions of Ono. In particular we show that for fixed ℓ and n, pℓ(a;n) roughly fits a given distribution that is dependent on ℓ, is centered near image and has width image. We also give an asymptotic formula for ∑a(−1)apℓ(a;n) that is valid whenever image is not, for any k, within a multiplicative factor of clogℓ of ℓk. This formula is of the form image where c and κ are specific functions of n and the sign is determined by n.