The number of distinct part sizes in a random integer partition

AbstractWe 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 Dn(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: Dn(λ) ⩽ An + xBn}P(n) → 12π ∫−∞xℓ−t22dt as n → ∞, where An = (√6/π)n12 and Bn = (ρ6/2π − √54/π3)12n14