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arXiv · 2203.09826

Proofs of some conjectures of Chan-Mao-Osburn on Beck's partition statistics

Liuxin Jin·Eric H. Liu·Ernest X. W. Xia
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Abstract

Recently, George Beck introduced two partition statistics $NT(m,j,n)$ and $M_ω(m,j,n)$, which denote the total number of parts in the partition of $n$ with rank congruent to $m$ modulo $j$ and the total number of ones in the partition of $n$ with crank congruent to $m$ modulo $j$, respectively. Andrews proved a congruence on $NT(m,5,n)$ which was conjectured by Beck. Very recently, Chan, Mao and Osburn established a number of Andrews-Beck type congruences and posed several conjectures involving $NT(m,j,n)$ and $M_ω(m,j,n)$. Some of those conjectures were proved by Chern and Mao. In this paper, we confirm the remainder three conjectures of Chan-Mao-Osburn and two conjectures due to Mao. We also present two new conjectures on $M_ω(m,j,n)$ and $NT(m,j,n)$.

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Liuxin Jin, Eric H. Liu, Ernest X. W. Xia. 2022-03-18. Proofs of some conjectures of Chan-Mao-Osburn on Beck's partition statistics. https://arxiv.org/abs/2203.09826

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