Multiranks for Partitions into Multi-Colors

Abstract

We generalize Hammond-Lewis birank to multiranks for partitions into colors and give combinatorial interpretations for multipartitions such as $b(n)$ defined by H. Zhao and Z. Zhong and $Q_{p_1;p_2}(n)$ defined by Toh congruences modulo 3, 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 ).…”

Rank and Crank Analogs for some Colored Partitions

“…About further research of arithmetic properties of cubic partitions, overcubic partitions and other colored partitions, some interesting results can be found in [18,21,23,24]. The first author [25] of the present paper generalized Hammond-Lewis birank and gave combinatorial interpretations for some colored partitions.…”

Arithmetic properties for cubic partition pairs modulo powers of 3

“…where i = 2, 3, 4, 6. For combinatorial proofs of the first two congruences, the reader may refer to Kim [5] and Zhou [9] respectively. In a recent paper, Lin [6] studied the arithmetic properties of b(n) modulo 27.…”