An almost‐bijective proof of an asymptotic property of partitions

Let 𝒫_n be the set of all distinct ordered pairs (λ,λ_i), where λ is a partition of n and λ_i is a part size of λ. The primary result of this note is a combinatorial proof that the probability that, for a pair (λ,λ_i) chosen uniformly at random from 𝒫_n, the multiplicity of λ_i in λ is 1 tends to 1/2 as n →∞. This is inspired by work of Corteel, Pittel, Savage, and Wilf (Random Structures and Algorithms 14 (1999), 185–197). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007