Enumeration of diagonally colored Young diagrams

TL;DR

This paper presents a new combinatorial proof for a formula counting diagonally colored Young diagrams, linking it to Nakajima quiver varieties and expressing the series as infinite products in special cases.

Contribution

It provides a direct combinatorial proof of the generating series formula, connecting Young diagrams, quiver varieties, and generalized Frobenius partitions.

Findings

Derived a closed-form generating series for diagonally colored Young diagrams

Connected the series to Euler characteristics of Nakajima quiver varieties

Expressed series as infinite products in specific cases

Abstract

In this note we give a new proof of a closed formula for the multivariable generating series of diagonally colored Young diagrams. This series also describes the Euler characteristics of certain Nakajima quiver varieties. Our proof is a direct combinatorial argument, based on Andrews' work on generalized Frobenius partitions. We also obtain representations of these series in some particular cases as infinite products.