The number of distinct part sizes in a random integer partition

Abstract

We prove a central limit theorem for the number of different part sizes in a random integer partition. If λ is one of the P( n) partitions of the integer n, let D n (λ) be the number of distinct part sizes that λ has. (Each part size counts once, even though there may be many parts of a given size.) For any fixed x, #(λ: D n(λ) ⩽ A n + xB n} P(n) → 1 2π ∫ −∞ x ℓ −t 2 2 dt as n → ∞, where A n = (√6/π)n 1 2 and B n = (ρ6/2π − √54/π 3) 1 2 n 1 4 .