The KOH terms and classes of unimodal N-modular diagrams

TL;DR

This paper provides a combinatorial interpretation of Zeilberger's KOH identity using modified N-modular diagrams of integer partitions, extending previous results and connecting to applications in quantum physics.

Contribution

It introduces a new combinatorial framework for understanding the KOH identity through modular diagrams, with novel bijections and extensions of recent theorems.

Findings

Established a combinatorial interpretation of KOH terms

Developed bijections for classes of modular diagrams

Extended theorems with applications to quantum physics

Abstract

We show how certain suitably modified N-modular diagrams of integer partitions provide a nice combinatorial interpretation for the general term of Zeilberger's KOH identity. This identity is the reformulation of O'Hara's famous proof of the unimodality of the Gaussian polynomial as a combinatorial identity. In particular, we determine, using different bijections, two main natural classes of modular diagrams of partitions with bounded parts and length, having the KOH terms as their generating functions. One of our results greatly extends recent theorems of J. Quinn et al., which presented striking applications to quantum physics.