Combinatorial Theory

In 2011, Han and Ji proved addition-multiplication theorems for integer partitions, from which they derived modular analogues of many classical identities involving hook-length. In the present paper, we prove addition-multiplication theorems for the subset of self-conjugate partitions. Although difficulties arise due to parity questions, we are almost always able to include the BG-rank introduced by Berkovich and Garvan. This gives us as consequences many self-conjugate modular versions of classical hook-lengths identities for partitions. Our tools are mainly based on fine properties of the Littlewood decomposition restricted to self-conjugate partitions.

Mathematics Subject Classifications: 05A15, 05A17, 05A19, 05E05, 05E10, 11P81

Keywords: Hook-length formulas, BGP-ranks, Integers partitions, Littlewood decomposition, core partitions