Proofs of some conjectures of Chan-Mao-Osburn on Beck's partition statistics (Q2150972)

Recently, George Beck introduced two partition statistics \(NT(m,j,n)\) and \(M_\omega(m,j,n)\) for integer partition, which denotes the total number of parts in all partitions of \(n\) with rank congruent to \(m\) modulo \(j\) and the total number of ones in all partitions of \(n\) with crank congruent to \(m\) modulo \(j\), respectively. He further conjectured the following congruences for \(NT(m,j,n)\): \[ \sum_{m=1}^4mNT(m,5,5n+1)\equiv\sum_{m=1}^4mNT(m,5,5n+4)\equiv0\pmod{5}\tag{1} \] and \[ \begin{multlined} NT(1,7,7n+i)+NT(2,7,7n+i)-NT(3,7,7n+i) \\ +NT(4,7,7n+i)-NT(5,7,7n+i)-NT(6,7,7n+i)\equiv0\pmod{7}, \end{multlined}\tag{2} \] where \(i\in\{1,5\}\) and \(n\geq0\). The conjectural congruences (1) and (2) were later confirmed by \textit{G. E. Andrews} [Int. J. Number Theory 17, No. 2, 239--249 (2021; Zbl 1465.11200)], and therefore called Andrews-Beck type congruences. Soon after, \textit{S. H. Chan}, \textit{R. Mao} and \textit{R. Osburn} [J. Math. Anal. Appl. 495, No. 2, Article ID 124771, 14 p. (2021; Zbl 1464.11106)] derived many Andrews-Beck type congruences and posed several conjectures involving \(NT(m,j,n)\) and \(M_\omega(m,j,n)\), some of which were confirmed by \textit{S. Chern} [Int. J. Number Theory 18, No. 1, 141--163 (2022; Zbl 1491.11095)] and \textit{R. Mao} [Ramanujan J. https://doi.org/10.1007/s11139-021-00485-w]. In this paper under review, the authors confirm the remaining three conjectures of Chan-Mao-Osburn and two conjectures due to Mao. More specifically, they proved that \[ \begin{multlined} \sum_{n=0}^\infty\big(NT(1,5,5n+4)-NT(4,5,5n+4) \\ +2M_\omega(2,5,5n+4)-2M_\omega(2,5,5n+4)\big)q^n=-5\prod_{k=1}^\infty\frac{(1-q^{5k})^4}{(1-q^k)}, \end{multlined}\tag{3} \] and for any \(n\geq0\), \begin{align*} M_\omega(2,5,5n+4)-M_\omega(3,5,5n+4) &=2NT(1,5,5n+4)-2NT(4,5,5n+4),\tag{4}\\ M_\omega(1,5,5n+4)-M_\omega(4,5,5n+4) &=2M_\omega(3,5,5n+4)-2M_\omega(2,5,5n+4),\tag{5}\\ M_\omega(1,5,5n+2)-M_\omega(4,5,5n+2) &=2NT(3,5,5n+2)-2NT(2,5,5n+2),\tag{6}\\ M_\omega(2,5,5n+1)-M_\omega(3,5,5n+1) &=NT(2,5,5n+1)-NT(3,5,5n+1).\tag{7} \end{align*} Moreover, Chan, Mao and Osburn posed the following conjectures: \begin{align*} \lim_{n\rightarrow\infty}\frac{\#\{k\mid M_\omega(i,5,k)\equiv M_\omega(j,5,k)\pmod{2},\;1\leq k\leq n\}}{n} &=\frac{1}{2},\\ \lim_{n\rightarrow\infty}\frac{\#\{k\mid M_\omega(1,5,k)\equiv M_\omega(4,5,k)\pmod{2},\;1\leq k\leq n\}}{n} &=\frac{3}{10},\\ \lim_{n\rightarrow\infty}\frac{\#\{k\mid M_\omega(2,5,k)\equiv M_\omega(3,5,k)\pmod{2},\;1\leq k\leq n\}}{n} &=\frac{2}{5} \end{align*} and \[ \lim_{n\rightarrow\infty}\frac{\#\{k\mid M_\omega(a,5,k)\equiv M_\omega(b,5,k)\pmod{2},\;1\leq k\leq n\}}{n} =\frac{1}{2}, \] where \(0\leq i<j\leq4\), \(i+j\neq5\) and \(0\leq a<b\leq4\).