On the Average Size of an $(\overline{s},\overline{t})$-Core Partition

Abstract

Let $s$ and $t$ be two coprime integers. Bessenrodt and Olsson obtained the number of $(\overline{s},\overline{t})$-cores for odd $s$ and odd $t$ by establishing a bijection between the lattice paths in $(s,t)$ Yin-Yang diagram and $(\overline{s},\overline{t})$-cores. In this paper, motivated by their results, we extend the definition of Yin-Yang diagram and the bijection to all possible coprime pairs $(s,t)$, then obtain that the number of $(\overline{s},\overline{t})$-cores is $\binom{\lfloor s/2 \rfloor + \lfloor t/2 \rfloor}{\lfloor s/2 \rfloor}$. Furthermore, based on the identities of Chen-Huang-Wang, we determine the average size of an $(\overline{s},\overline{t})$-core depending on the parity of $s$, which is $(s-1) (t-1) (s+t-2)/48$ if $s$ and $t$ are both odd, or $(t-1) (s^2+st-3s+2t+2)/48$ if $s$ is even and $t$ is odd.