Combinatorial construction of known positive series for partition classes defined by Capparelli, Meurman, Primc, and Primc in the $k$=1 Case

TL;DR

This paper develops combinatorial methods to construct positive series for partition classes related to CMPP partitions, extending previous series and providing combinatorial interpretations for several cases.

Contribution

It introduces a combinatorial construction of positive series for CMPP partitions, extending known series and interpreting Russell's bivariate series.

Findings

Extended series for CMPP partitions with combinatorial interpretations.

Connected Russell's bivariate series to base partitions and moves.

Supplied missing cases of positive series for partition classes.

Abstract

Recently, Capparelli, Meurman, A. Primc and M. Primc introduced a class of colored partitions which has since been called CMPP partitions. This generalized earlier work by M. Primc and \v{S}iki\'{c}, and by Trup\v{c}evi\'{c}. One main reason why CMPP partitions are significant is the authors' conjecture that the generating functions are infinite products in all cases. CMPP partitions are true extensions of the partition classes in the Rogers-Ramanujan-Gordon identities which are defined by difference conditions. As such, a natural question is to look for generating functions similar to the series side of Andrews-Gordon identities. Russell found such bivariate series for one case. These evidently positive series overlap with the positive series found earlier by Griffin, Ono and Warnaar in the edge cases. Russell used symbolic computation in the proofs. We will combinatorially interpret Russell's bivariate series extending one case of the series due to Griffin, Ono and Warnaar in a base partition and moves setting, and supply some missing cases, as well.