Stanton type inequality for s-asymmetric t-core partitions

Abstract

In 1999, Stanton conjectured that if t≥4 and n≠t+1, then ct(n)≤ct+1(n), where ct(n) is the number of t -core partitions of n . This conjecture has since been proven. An analogous result exists for self-conjugate t -core partitions: for t≥9 or t=6,8 and n sufficiently large, it follows that sct(n)<sct+2(n), where sct(n) is the number of self-conjugate t -core partitions of n . In this article, we extend these results to a more general class of partitions. Let cs,t(n) be the number of s -asymmetric t -core partitions of n . Using the circle method, we derive an asymptotic formula for cs,t(n) for s≥0 and t−s≥10. This asymptotic formula is then applied to prove that cs,t(n)≤cs,t+2(n) for sufficiently large n . Furthermore, we show that if t−s is odd, the inequality cs,t(n)≤cs,t+1(n) holds for n≫0. © 2026 Elsevier Inc.