A proof of the Auluck-Chowla-Gupta Conjecture

Abstract

Let p(n,k) denote the number of partitions of n into exactly k parts. In 1942 Auluck, Chowla and Gupta conjectured that for every n the sequence p(n,1), p(n,2), ..., p(n,n) is unimodal. Szekeres proved this for all sufficiently large n, but with a threshold that was never made explicit. We prove the conjecture for every n>= 1. For n>= 10^5 we give an analytic argument with fully explicit constants: an exact probabilistic representation of the ratio p(n,k+1)/p(n,k), a second-order saddle-point main term with a rigorous uniform error bound in the transition window, and separate elementary or saddle-point estimates in the two tails. Every numerical constant in this argument is certified in ball arithmetic. For n<= 2*10^5 the conjecture is verified exhaustively by two independent programs, one of them using exact integer arithmetic only. Since log-concavity of k ->p(n,k) fails for every n>= 4, the argument works with first differences and never uses log-concavity.