Capparelli's partition theorem as part of an infinite hierarchy: Combinatorial and Weighted Words extensions of recent work

Abstract

In a recent paper, the authors introduced an infinite hierarchy of $q$-hypergeometric identities, of which the first three orders, $0$, $1$, and $2$, relate to the partition theorems of Euler, Lebesgue, and Capparelli, and stated a partition theorem at order 4 which lies beyond Capparelli's theorem. Here, we first state certain partition theorems that hold at all even orders beyond Capparelli and provide bijective proofs for these theorems. In doing so, we show that there is a fourfold infinite hierarchy of partition theorems that emanates from Capparelli's theorem, which is the base case. It is also shown that the equality of two of the four generating functions holds for all orders, odd and even. Lastly, a very general framework for the remaining two functions is constructed via the method of weighted words, encompassing all possible orders and yielding several infinite hierarchies with different dilations and translations.