New partition identities for odd w odd

TL;DR

This paper conjectures new Rogers-Ramanujan type partition identities involving arrays with an odd number of rows, where the first and last rows contain even positive integers, supported by numerical evidence.

Contribution

It introduces novel conjectures for partition identities with specific parity conditions, extending previous work related to affine Lie algebra representations.

Findings

Numerical evidence supports the conjectured identities.

Identifies a new class of partition identities with even boundary rows.

Differences from previous odd-boundary partition identities are highlighted.

Abstract

In this note we conjecture Rogers-Ramanujan type colored partition identities for an array with odd number of rows w such that the first and the last row consist of even positive integers. In a strange way this is different from the partition identities for the array with odd number of rows w such that the first and the last row consist of odd positive integers -- the partition identities conjectured by S. Capparelli, A. Meurman, A. Primc and the author and related to standard representations of the affine Lie algebra of type $C^{(1)}_\ell$ for $w=2\ell+1$. The conjecture is based on numerical evidence.