Some inequalities and equalities on Lin-Peng-Toh's partition statistic for \(k\)-colored partitions (Q6619532)

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 numbers \(\lambda_i\) are called the parts of \(\lambda\). \textit{F. J. Dyson} [Eureka (Cambridge), 8, 10--15 (1944)] defined the rank of a partition \(\lambda\), which is the largest part of the partition minus the number of parts, to provide a combinatorial interpretation of Ramanujan's congruences on \(p(n)\). He conjectured that this partition statistic can interpret Ramanujan's congruences modulo \(5\) and \(7\) combinatorially. However, it can not give a combinatorial interpretation for Ramanujan's congruence modulo \(11\). 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)].\N\NRecently, George Beck defined 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,\N\begin{align*}\NNT(r,m,n) &:=\sum_{\substack{\lambda\vdash n\\\N\textrm{rank}(\lambda)\equiv r\pmod{m}}}\#(\lambda),\\\NM_\omega(r,m,n) &:=\sum_{\substack{\lambda\vdash n\\\N\textrm{crank}(\lambda)\equiv r\pmod{m}}}\omega(\lambda),\N\end{align*}\Nwhere \(\#(\lambda)\) and \(\omega(\lambda)\) denote the number of parts of \(\lambda\) and the number of ones in \(\lambda\). Beck posed the following conjectural congruences:\N\begin{align*}\N&NT(1,5,5n+i)+2NT(2,5,5n+i) \\\N&\qquad -NT(3,5,5n+i)-NT(4,5,5n+i)\equiv0\pmod{5},\tag{1}\\\N&NT(1,7,7n+j)+NT(2,7,7n+j)\\\N&\qquad -NT(3,7,7n+j)+NT(4,7,7n+j) \\\N&\qquad -NT(5,7,7n+j) -NT(6,7,7n+j)\equiv0\pmod{7},\tag{2}\N\end{align*}\Nwhere \(i\in\{1,4\}\) and \(j\in\{1,5\}\). The congruences \((1)\) and \((2)\) were confirmed by \textit{G. E. Andrews} [Int. J. Number Theory 17, No. 2, 239--249 (2021; Zbl 1465.11200)].\N\NMotivated by these work, \textit{B. L. S. Lin}, \textit{L. Peng} and \textit{P. Toh} [Discrete Math. 344, No. 8, Article ID 112450, 13 p. (2021; Zbl 1478.11122)] considered the generalized crank for \(k\)-colored partitions, which was defined by \textit{S. Fu} and \textit{D. Tang} [J. Number Theory 184, 485--497 (2018; Zbl 1420.11130)], where \(k\geq2\). A \(k\)-colored partition \(\lambda\) of a positive integer \(n\) is a \(k\)-tuple of partitions \(\lambda:=\big(\lambda^{(1)},\lambda^{(2)},\ldots, \lambda^{(k)}\big)\) such that \(|\lambda^{(1)}|+|\lambda^{(2)}|+\cdots +|\lambda^{(k)}|=n\). If \(\lambda\) is a \(k\)-colored partition of \(n\), we denote it by \(\lambda\vdash n\). Fu and Tang defined a generalized crank for \(k\)-colored partitions by\N\[\N\textrm{crank}_k(\lambda)=\#\big(\lambda^{(1)}\big) -\#\big(\lambda^{(2)}\big).\N\]\NLet \(r\), \(m\), \(n\), \(k\) be integers with \(m\geq1\), \(k\geq2\), \(n\geq1\) and \(0\leq r\leq m-1\). Define\N\[\NN\!B_k(r,m,n):=\sum_{\substack{\lambda\vdash n\\\N\textrm{crank}_k(\lambda) \equiv r\pmod{m}}}\#\big(\lambda^{(1)}\big).\N\]\NLin, Peng, and Toh proved many Andrews--Beck type congruences for \(N\!B_k(r,m,n)\).\N\NIn this paper, the authors first establish the generating functions for \(N\!B_k(r,m,n)\) when \(m\in\{2,3,4\}\). For example, they prove that for any \(k\geq2\),\N\begin{align*}\N\sum_{n=0}^\infty N\!B_k(0,4,n)q^n &=\dfrac{1}{4(q;q)_\infty^k}\sum_{n=1}^\infty \dfrac{q^n}{1-q^n}-\dfrac{(q;q)_\infty^{2-k}(q^2;q^2)_\infty}{2(q^4;q^4)_\infty} \sum_{n=1}^\infty\dfrac{q^{2n}}{1+q^{2n}}\\\N&\quad+\dfrac{(q;q)_\infty^{4-k}}{4(q^2;q^2)_\infty^2}\sum_{n=1}^\infty \dfrac{q^n}{1+q^n},\\\N\sum_{n=0}^\infty N\!B_k(1,4,n)q^n &=\dfrac{1}{4(q;q)_\infty^k}\sum_{n=1}^\infty \dfrac{q^n}{1-q^n}+\dfrac{(q;q)_\infty^{2-k}(q^2;q^2)_\infty}{2(q^4;q^4)_\infty} \sum_{n=1}^\infty\dfrac{q^n}{1+q^{2n}}\\\N&\quad+\dfrac{(q;q)_\infty^{4-k}}{4(q^2;q^2)_\infty^2}\sum_{n=1}^\infty \dfrac{q^n}{1+q^n},\\\N\sum_{n=0}^\infty N\!B_k(2,4,n)q^n &=\dfrac{1}{4(q;q)_\infty^k}\sum_{n=1}^\infty \dfrac{q^n}{1-q^n}+\dfrac{(q;q)_\infty^{2-k}(q^2;q^2)_\infty}{2(q^4;q^4)_\infty} \sum_{n=1}^\infty\dfrac{q^{2n}}{1+q^{2n}}\\\N&\quad-\dfrac{(q;q)_\infty^{4-k}}{4(q^2;q^2)_\infty^2}\sum_{n=1}^\infty \dfrac{q^n}{1+q^n},\\\N\sum_{n=0}^\infty N\!B_k(3,4,n)q^n &=\dfrac{1}{4(q;q)_\infty^k}\sum_{n=1}^\infty \dfrac{q^n}{1-q^n}-\dfrac{(q;q)_\infty^{2-k}(q^2;q^2)_\infty}{2(q^4;q^4)_\infty} \sum_{n=1}^\infty\dfrac{q^n}{1+q^{2n}}\\\N&\quad+\dfrac{(q;q)_\infty^{4-k}}{4(q^2;q^2)_\infty^2}\sum_{n=1}^\infty \dfrac{q^n}{1+q^n},\N\end{align*}\Nwhere we adopt the following notation:\N\begin{align*}\N(a;q)_\infty:=\prod_{j=0}^\infty(1-aq^j),\qquad|q|<1.\N\end{align*}\NAs an immediate consequence, the authors prove many inequalities and equalities for \(N\!B_k(r,m,n)\). For instance, the authors prove that for any \(n\geq1\),\N\begin{align*}\NN\!B_k(1,2,n) &>N\!B_k(0,2,n),\,\,\,\,\,\,\text{if}\,\,k\geq5,\\\NN\!B_k(3,4,n) &>N\!B_k(1,4,n),\,\,\,\,\,\,\text{if}\,\,k\geq3\N\end{align*}\Nand\N\[\NN\!B_3(2,3,3n-1)=N\!B_3(1,3,3n-1)>N\!B_3(0,3,3n-1).\N\]