Method of Weighted Words on Cylindric Partitions

TL;DR

This paper develops a combinatorial method using weighted words to derive alternative generating functions for cylindric partitions with specific profiles and ranks, expanding on previous formulas and techniques.

Contribution

It adapts the method of weighted words to cylindric partitions, providing new combinatorial expressions for their generating functions beyond Borodin's formula.

Findings

Derived alternative generating functions for cylindric partitions

Extended the method of weighted words to new combinatorial contexts

Provided explicit formulas for profiles with ranks 2 and levels 2, 3, 4

Abstract

We study the generating functions of cylindric partitions having profile $c=(c_1, c_2, \ldots, c_r)$ with rank $2$ and levels $2, 3$ and $4$. As a result, we give expressions alternative to Borodin's formula for these generating functions. We use the method of weighted words which was first introduced by Alladi and Gordon, later was applied by Dousse in a new version to prove some partition identities and to get infinite products. We adapt the method to our subject with a more combinatorial approach.