A Note on the Exact Expected Length of the Kth Part of a Random Partition

Kessler and Livingstone proved an asymptotic formula for the expected length of the largest part of a partition drawn uniformly at random. As a first step they gave an exact formula expressed as a weighted sum of Euler’s partition function. Here we give a short bijective proof of a generalization of this exact formula to the expected length of the kth part. 1. Results By λ ! n we will mean that λ is a partition of n. This means that λ is a finite non-increasing sequence of positive integers, λ1 ≥ · · · ≥ λN > 0, which sums to n. The number of partitions of n is Euler’s famous partition function p(n), with p(0) = 1 by convention. Corteel et al. [1] mention a well-known partition identity attributed to Stanley: The expected number of different part sizes of a uniformly drawn partition λ ! n is 1 p(n) ∑ !≥1 " · pδ(n, ") = 1 p(n) n−1 ∑