A limit theorem concerning the likely shape of the Ferrers diagram

Abstract

We study the asymptotic behaviour of the 5th largest part L s , n in a random partition of a positive integer n as n ->· «>. The weak convergence of the distribution of L s , n to the Gaussian distribution is established provided s is of order w 1 / 2 and n ->· °°.

On the Maximal Multiplicity of Parts in a Random Integer Partition

“…The next lemma, whose proof may be found in [14], suggests a tool that we shall subsequently use in Section 3 to obtain the main term in our asymptotics.…”