On the moments of normal distributions and numbers of standard Young tableaux

The exponential generating function of the number tau(n) of standard Young tableaux of size n (or the number of involutions of n letters) is known to be e (+)t(2)/2 , which coincides with the moment generating function of normal distribution N(1, 1). We apply Laplace's method to the moment integral to derive a new asymptotic formula of tau(n) which is better than the known result. Meanwhile, by using of Can and Joyce's result in 2012, the number S-n, of skew product-type standard Young tableaux of size n is derived to be the n-th moment of normal distribution NO-, tau), where r is geometric random variable. The generating function and the asymptotic formula of S-n, are obtained. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The study focused on the number of standard Young tableaux and skew product-type standard Young tableaux, deriving their respective generating functions and asymptotic formulas.