Some inequalities and equalities on Lin–Peng–Toh’s partition statistic for k-colored partitions

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发表日期：

2024年8月12日

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Yueya Hu,
Eric H. Liu,
Olivia X. M. Yao

摘要

A k-colored partition $\pi $ $π$ of a positive integer n is a k-tuple of partitions $\pi =(\pi ^{(1)},\ldots , \pi ^{(k)})$ $π = (π^{(1)}, \dots, π^{(k)})$ such that $|\pi ^{(1)}| +\cdots +|\pi ^{(k)}|=n$ $| π^{(1)} | + \dots + | π^{(k)} | = n$ . Recently, Fu and Tang defined a generalized crank for k-colored partitions by $ \textrm{crank}_k(\pi ) =\#(\pi ^{(1)})-\#(\pi ^{(2)}) $ ${crank}_{k} (π) = # (π^{(1)}) - # (π^{(2)})$ , where $\#(\pi ^{(i)})$ $# (π^{(i)})$ denotes the number of parts in $\pi ^{(i)}$ $π^{(i)}$ . They also proved some inequalities and equalities for $M_k(m,j,n)$ $M_{k} (m, j, n)$ which counts the number of k-colored partitions of n with generalized crank congruent to m modulo j. Very recently, Lin, Peng and Toh established some new Andrews–Beck type congruences on $NB_k(m,j,n)$ $N B_{k} (m, j, n)$ which denotes the total number of parts of $\pi ^{(1)}$ $π^{(1)}$ in each k-colored partition $\pi $ $π$ of n with $ \textrm{crank}_k(\pi )$ ${crank}_{k} (π)$ congruent to m modulo j. In this paper, motivated by the work of Fu–Tang and Lin–Peng–Toh, we establish the generating functions for $NB_k(m,j,n)$ $N B_{k} (m, j, n)$ when $j=2,3,4$ $j = 2, 3, 4$ and deduce some new inequalities and equalities for $NB_k(m,j,n)$ $N B_{k} (m, j, n)$ .