Rank and Crank Analogs for some Colored Partitions

Abstract

We establish some rank and crank analogs for partitions into distinct colors and give combinatorial interpretations for colored partitions such as partitions defined by Toh, Zhang and Wang congruences modulo 5, 7.

Multiranks and classical theta functions

“…• First it should be acknowledged that in the work of Zhou [44], Zhou and Zhang [45], they used exactly the same multirank as r 4 to explain congruences mod 5 for 4colored partitions of type (1, 2), (1, 3) and (2, 3) respectively, so our Theorem 3.6 below can be viewed as a natural complement to their work and shows the full power of r 4 . • Secondly, this definition of r 4 entitles us to consider a natural rank-preserving bijection between P [c,d] and P [d,c] : (π r , π b , π y , π o ) → (π y , π o , π r , π b ).…”

“…For subsequent studies either directly involving a(n) or encouraged by the success of a(n), see for example [7,13,14,17,32,33,39,40,[42][43][44][45]. In particular, we would like to mention the work of Zhao and Zhong [43] on the so-called "cubic partition pairs" b(n), whose generating function is given as…”