Andrews-Beck type congruences modulo 2 and 4 for Beck's partition statistics (Q6136243)

A partition \(\lambda\) of a nonnegative integer \(n\) is a sequence of positive integers \(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_k>0\) such that \(\sum_{i=1}^k\lambda_i=n\). The \(\lambda_i\) for \(1\leq i\leq k\) are called the parts of \(\lambda\). To give a combinatorial interpretation of Ramanujan's celebrated congruences on \(p(n)\), \textit{F. J. Dyson} [Eureka 8, 10--15 (1944)] defined the rank of a partition \(\lambda\), which is the largest part of the partition minus the number of parts. He conjectured that this partition statistic can interpret Ramanujan's congruences modulo \(5\) and \(7\) combinatorially. However, it can not provide a combinatorial interpretation for Ramanujan's congruence modulo \(11\). Therefore, Dyson further conjectured that there exists another partition statistic that he named ``crank'', which can interpret Ramanujan's congruence modulo \(11\) combinatorially. In 1988, this partition statistic was eventually discovered by \textit{G .E. Andrews} and \textit{F. G. Garvan} [Bull. Am. Math. Soc., New Ser. 18, No. 2, 167--171 (1988; Zbl 0646.10008)]. Recently, G. Beck introduced two partition statistics \(NT(r,m,n)\) and \(M_\omega(r,m,n)\), which denote the total number of parts in the partitions of \(n\) with rank congruent to \(r\) modulo \(m\) and the total number of ones in the partitions of \(n\) with crank congruent to \(r\) modulo \(m\), respectively, that is, \begin{align*} NT(r,m,n) &:=\sum_{\substack{\lambda\vdash n\\ \operatorname{rank}(\lambda)\equiv r(\operatorname{mod} m)}} \#(\lambda),\\ M_\omega(r,m,n) &:=\sum_{\substack{\lambda\vdash n\\ \operatorname{crank}(\lambda)\equiv r(\operatorname{mod} m)}} \omega(\lambda), \end{align*} where \(\#(\lambda)\) and \(\omega(\lambda)\) denote the number of parts of \(\lambda\) and the number of ones in \(\lambda\). Beck posed the following conjectural congruences: \[ NT(1,5,5n+i)+2NT(2,5,5n+i) -NT(3,5,5n+i)-NT(4,5,5n+i)\equiv0\pmod{5},\tag{1} \] \[ \begin{multlined} NT(1,7,7n+j)+NT(2,7,7n+j) -NT(3,7,7n+j)+NT(4,7,7n+j)\\ -NT(5,7,7n+j)-NT(6,7,7n+j)\equiv0\pmod{7}, \end{multlined}\tag{2} \] where \(i\in\{1,4\}\) and \(j\in\{1,5\}\). The 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)]. The purpose of this paper is to establish some Andrews-Beck type congruences modulo \(2\) and \(4\) for \(NT(r,m,n)\) and \(M_\omega(r,m,n)\). For example, the authors prove that for any \(k\geq1\) and \(n\geq0\), \begin{align*} NT(2k,4k,n)\equiv NT(2k,4k,n) &\equiv0\pmod{2},\\ M_\omega(1,8,4n)-M_\omega(7,8,4n)\equiv M_\omega(3,8,4n)-M_\omega(5,8,4n) &\equiv0\pmod{2},\\ 2M_\omega(1,8,4n+3)+M_\omega(2,8,4n+3) -M_\omega(6,8,4n+3)-2M_\omega(7,8,4n+3) &\equiv0\pmod{4}. \end{align*} Moreover, they proved that for any \(n\geq0\), \begin{align*} M_\omega(b,k,n) &\equiv M_\omega(k-b,k,n)\pmod{\gcd(k,b)},\\ NT(b,k,n) &\equiv NT(k-b,k,n)\pmod{\gcd(k,b)}, \end{align*} where \(k\), \(b\) are integers with \(0\leq b\leq k-1\). The main technique is the use of classical \(q\)-series and their clever manipulation.