Tight cylindric partitions

TL;DR

This paper introduces generating functions for tight cylindric partitions, providing new functional equations, closed-form formulas for 2-rowed cases, and establishing bijections with previously studied partitions.

Contribution

It develops the first generating function framework for tight cylindric partitions, including functional equations and explicit formulas for specific cases.

Findings

Derived analogs of Corteel--Welsh functional equations for tight cylindric partitions.

Obtained closed-form bivariate generating functions for 2-rowed cases.

Established a bijection with a class of partitions studied by Dousse, Hardiman, and Konan.

Abstract

In this note, we initiate the study of generating functions for tight cylindric partitions. For general (i.e., $r$-rowed for $r\geq 2$) tight cylindric partitions, we provide analogs of the Corteel--Welsh functional equations. We prove closed forms for the bivariate generating functions for 2-rowed tight cylindric partitions. We also show that these partitions are in bijection with a class of partitions studied by Dousse, Hardiman, and Konan.