MacMahon's partition analysis. XIV: Partitions with \(n\) copies of \(n\) (Q6187336)

In this paper, the authors apply the method of partition analysis to partitions with \(n\) copies of \(n\) and also partitions with \((n+1)\) copies of \(n\), which are subscripted positive integers wherein the subscript does not exceed the integer. We order these objects lexicographically \[ 1_1<2_1<2_2<3_1<3_2<3_3<4_1<4_2<4_3<4_4<\cdots. \] We now define the weighted difference between \(m_i\) and \(n_j\) to be \[ ((m_i-n_j)):=m-n-i-j. \] Additionally, we shall sometimes consider partitions with \((n+1)\) copies of \(n\). Here the subscript 0 will be allowed, and lexicographic order is maintained; i.e., \[ 1_0<1_1<2_0<2_1<2_2<3_0<3_1<3_2<3_3<\cdots. \] The study of partitions with \(n\) copies of \(n\) had its origins in Regime III of the hard hexagon model [\textit{G. E. Andrews} et al., J. Stat. Phys. 35, 193--266 (1984; Zbl 0589.60093)]. As an immediate application, the authors derive the following multivariable generating functions related to classical Rogers-Ramanujan type identities: Theorem. The generating function for partitions with \(n\) copies of \(n\) with \(m\) parts in which the weighted difference between parts greater than \(r-1\) with \(r\geq-2\) is given by \[ \dfrac{x_1^{(m-1)r+(2m-1)}x_2^{(m-2)r+(2m-3)}\cdots x_m^{0\cdot r+1}}{\prod_{i=1}^m(1-x_1x_2\cdots x_i)(1-x_1) \prod_{i=2}^m\big(1-x_1^2x_2^2\cdots x_{i-1}^2x_i\big)}, \] where the exponent of \(x_i\) accounts for the \(i\)th part of the partition in question. Moreover, partitions with \(n\) copies of \(n\) are extended to partition diamonds yielding numerous new results including a natural connection to overpartitions and a variety of partition congruences. For example, let \(\textrm{PDN1}(n)\) denote the number of partition diamonds of \(n\) with some conditions. The authors first prove \[ \sum_{n=0}^\infty\textrm{PDN1}(n)q^n=\prod_{n=1}^\infty \dfrac{(1+q^n)^2}{(1-q^n)^3}.\tag{1} \] Utilizing the package \(\texttt{RaduRK}\) due to \textit{N. A. Smoot} [J. Symb. Comput. 104, 276--311 (2021; Zbl 1462.11093)], they obtained several congruences modulo small powers of 5 and 7 for \(\textrm{PDN1}(n)\), namely, \begin{align*} \textrm{PDN1}(25n+24) &\equiv0\pmod{5},\tag{2}\\ \textrm{PDN1}(125n+74) &\equiv0\pmod{25},\tag{3}\\ \textrm{PDN1}(125n+124) &\equiv0\pmod{25},\tag{4}\\ \textrm{PDN1}(7n+5) &\equiv0\pmod{7},\tag{5}\\ \textrm{PDN1}(49n+47) &\equiv0\pmod{49}. \end{align*} Interestingly, \textit{B. L. S. Lin} and \textit{A. Y. Z. Wang} [Colloq. Math. 154, No. 1, 137--148 (2018; Zbl 1429.11190)] also considered the partition function (1) in a purely analytic study of multiplicative inverses of identities of Ramanujan and Gordon. Actually, they provided an elementary proof of (2) and (5), and conjectured (3) and (4) with the help of a computer. These two congruences was proved by \textit{M. Bian}, \textit{D. Tang}, \textit{E. X. W. Xia} and \textit{F. Xue} [Ramanujan J. 55, No. 2, 497--515 (2021; Zbl 1491.11094)].