Taiwanese Journal of Mathematics

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

Joseph L. P. Wang and Jane Y. X. Yang

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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.

Article information

Taiwanese J. Math., Advance publication (2019), 16 pages.

First available in Project Euclid: 22 October 2018

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Primary: 05A17: Partitions of integers [See also 11P81, 11P82, 11P83] 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]

$(\overline{s},\overline{t})$-core partition Yin-Yang diagram lattice path average size


Wang, Joseph L. P.; Yang, Jane Y. X. On the Average Size of an $(\overline{s},\overline{t})$-Core Partition. Taiwanese J. Math., advance publication, 22 October 2018. doi:10.11650/tjm/181006. https://projecteuclid.org/euclid.twjm/1540195382

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