Rectangular analogues of the square paths conjecture and the univariate Delta conjecture

TL;DR

This paper introduces rectangular analogues of the square paths and Delta conjectures in algebraic combinatorics, extending existing conjectures to rectangular shapes and proving a special case when sides are coprime.

Contribution

It formulates a rectangular analogue of the square paths conjecture and proposes a combinatorial framework for a rectangular extension of the Delta conjecture, including a proof for coprime sides.

Findings

Proposed a rectangular analogue of the square paths conjecture.

Described combinatorial objects and statistics for rectangular Delta conjecture.

Proved the rectangular paths conjecture when rectangle sides are coprime.

Abstract

In this paper, we extend the rectangular side of the shuffle conjecture by stating a rectangular analogue of the square paths conjecture. In addition, we describe a set of combinatorial objects and one statistic that are a first step towards a rectangular extension of (the rise version of) the Delta conjecture, and of (the rise version of) the Delta square conjecture, corresponding to the case $q=1$ of an expected general statement. We also prove our new rectangular paths conjecture in the special case when the sides of the rectangle are coprime.