Armstrong's Conjecture for $(k,mk+1)$-Core Partitions

Abstract

A conjecture of Armstrong states that if $\gcd (a,b)=1$, then the average size of an $(a,b)$-core partition is $(a-1)(b-1)(a+b+1)/24$. Recently, Stanley and Zanello used a recursive argument to verify this conjecture when $a=b-1$. In this paper we use a variant of their method to establish Armstrong's conjecture in the more general setting where $a$ divides $b-1$.

arXiv comment17 pages, 3 figures

References in corpus (1)

• The t-core of an s-core

Cited by in corpus (7)

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• Numerical Sets, Core Partitions, and Integer Points in Polytopes
• Sizes of Simultaneous Core Partitions
• On the number of simultaneous core partitions with $d$-distinct parts
• Symmetry in maximal $(s-1,s+1)$ cores
• $(s,t)$-cores: a weighted version of Armstrong's conjecture
• On self-conjugate $(s,s+1,\dots ,s+k)$-core partitions