Combinatorial Theory

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.

Mathematics Subject Classifications: 05E05

Keywords: Macdonald polynomials, symmetric functions