Expected lengths and distribution functions for Young diagrams in the hook

We consider β-Plancherel measures [J. Baik, E. Rains, The asymptotics of monotone subsequences of involutions, Duke Math. J. 109 (2001) 205–281] on subsets of partitions—and their asymptotics. These subsets are the Young diagrams contained in a (k,ℓ)-hook, and we calculate the asymptotics of the expected shape of these diagrams, relative to such measures. We also calculate the asymptotics of the distribution function of the lengths of the rows and the columns for these diagrams. This might be considered as the restriction to the (k,ℓ)-hook of the fundamental work of Baik, Deift and Johansson [J. Baik, P. Deift, K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999) 1119–1178]. The above asymptotics are given here by ratios of certain Selberg-type multi-integrals.