A compositional shuffle conjecture specifying touch points of the Dyck path

TL;DR

This paper introduces a new $q,t$-enumeration of Dyck paths with specific touch point restrictions, conjectures its relation to Macdonald theory, and refines existing conjectures on diagonal harmonics and the shuffle conjecture.

Contribution

It proposes a refined conjecture linking Dyck path enumeration to Macdonald operator actions and highlights the role of generalized Hall-Littlewood polynomials in $q,t$-Catalan sequences.

Findings

Proposes a $q,t$-enumeration of Dyck paths with touch point constraints.

Conjectures a connection between this enumeration and the Macdonald $ abla$ operator.

Identifies generalized Hall-Littlewood polynomials as key components in $q,t$-Catalan theory.

Abstract

We introduce a $q,t$-enumeration of Dyck paths which are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory $\nabla$ operator applied to a Hall-Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the "shuffle conjecture" (Duke J. Math. $\mathbf {126}$ (2005), pp. 195-232) for $\nabla e_n[X]$. We bring to light that certain generalized Hall-Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of $q,t$-Catalan sequences and we prove a number of identities involving these functions.