{\(n\)}-color partition theoretic interpretations of some mock theta functions (Q1773158)

Let an \(n\)-color partition of a positive integer \(v\) be a partition in which a part of size \(n\) can come in \(n\)th different colors denoted by \(n_1,n_2, \dots, n_n\) and the parts satisfy the order \[ 1_1<2_1<2_2<3_1<3_2<\dots. \] For example the \(n\)-color partitions of 2 are \(2_1, 2_2, 1_1+1_1\). The weighted difference of two parts \(m_i\), \(n_j\), \(m\geq n\) is defined by \(m-n-i-j\). In this article, the author relates the coefficients of \(q^v\) of the series expansions of Ramanujan's mock theta functions to partition functions defined using \(n\)-color partitions. For example, he shows that if \(A(v)\) is the number of \(n\)-color partitions of \(v\) such that the even parts appear with even subscripts and odd with odd, for some \(k\), \(k_k\) is a part, and the weighted difference of any two consecutive parts is 0. Then \[ \sum_{v\geq 1} A(v) q^v= \sum_{m\geq 1}\frac{q^{m^2}} {(1-q)(1-q^3)\cdots (1-q^{2m-1})}. \]