Colored partitions and the hooklength formula: partition statistic identities

TL;DR

This paper explores new relations between hook lengths, frequencies, and part sizes in partitions, extending existing formulas through the study of colored partitions and their truncations, revealing identities at various polynomial degrees.

Contribution

It introduces novel relations between joint distributions in partitions by analyzing truncations of the hooklength formula and colored partitions, unifying concepts like k-colored partitions and overpartitions.

Findings

Established relations at constant and linear terms for all n

Derived identities for j=2 in quadratic terms

Extended the hooklength formula to colored partitions

Abstract

We give relations between the joint distributions of multiple hook lengths and of frequencies and part sizes in partitions, extending prior work in this area. These results are discovered by investigating truncations of the Han/Nekrasov-Okounkov hooklength formula and of (k,j)-colored partitions, a unification of k-colored partitions and overpartitions. We establish the observed relations at the constant and linear terms for all n, and for j=2 in their quadratic term, with the associated hook/frequency identities. Further results of this type seem likely.