Proof of the Alder-Andrews conjecture

Abstract

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 ±1(mod d + 3), respectively. He conjectured that q d (n) ≥ Q d (n). Andrews and Yee proved the conjecture for d = 2 s − 1 and also for d ≥ 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.

Generalizations of Alder's Conjecture via a Conjecture of Kang and Park

“…1 The d = 2 case is simply the second Rogers-Ramanujan identity. Finally, we investigate and generalize asymptotic results of Andrews [4], as well as Alfes, Jameson, and Lemke Oliver [3]. Theorem 1.9.…”

“…In 2004 and 2008, Yee [13,12] proved the conjecture for n ≥ 1, d ≥ 32, and d = 7. In 2011, Alfes, Jameson, and Lemke Oliver [3] proved Alder's Conjecture for n ≥ 1 and 4 ≤ d ≤ 30, d = 7, 15, thus completely resolving the conjecture.…”

“…In Section 5, we modify methods from the proof Theorem 1.3 to prove Theorem 1.6, adapt methods of Andrews [4] to prove Theorem 1.7, and generalize methods of Yee [13] to prove Theorem 1.8. In Section 6, we prove Theorem 1.9 as well as analog a method of Alfes, Jameson, and Lemke Oliver [3] to obtain an asymptotic expression for ∆ (a) d (n) with explicit error term. We then discuss how asymptotics could be useful to prove remaining cases of Conjecture 1.2.…”

Biases in Integer Partitions

“…Alder's conjecture (now proven by Andrews [3], Yee [14] and Alfes et al [2]) concerns inequalities between the number of partitions satisfying gap conditions and the number of partitions satisfying congruence conditions. In contrast, the Ehrenpreis problem concerns inequalities between two classes of partitions satisfying congruence conditions.…”

On a Conjecture of Andrews

“…G.E. Andrews considered a different generalization by considering the functions At a 2009 conference in Ottawa, he made the following conjecture 1 to accompany Alder's Conjecture (for more information on Alder's Conjecture, see [2], [3], [4], [7], and [8]).…”