Proof of the Alder-Andrews conjecture

Abstract:

Motivated by classical identities of Euler, Schur, and Rogers and Ramanujan, Alder investigated $q_d(n)$ and $Q_d(n),$ the number of partitions of $n$ into $d$-distinct parts and into parts which are $\pm 1 (\operatorname {mod}d+3)$, respectively. He conjectured that $q_d(n) \geq Q_d(n).$ Andrews and Yee proved the conjecture for $d = 2^s-1$ and also for $d \geq 32.$ We complete the proof of Andrews’s refinement of Alder’s conjecture by determining effective asymptotic estimates for these partition functions (correcting and refining earlier work of Meinardus), thereby reducing the conjecture to a finite computation.