A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

Abstract

Abstract. We introduce a q, t-enumeration of Dyck paths that 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 ∇ operator applied to a HallLittlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the "shuffle conjecture" (Duke J. Math. 126 (2005), pp. 195 − 232) for ∇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.

Compositional (km,kn)-Shuffle Conjectures

“…However, in the fall of 2008, Haglund, Morse, and Zabrocki [16] made a discovery that is nothing short of spectacular. They discovered that two slight deformations C a and B a of the well-known Hall-Littlewood operators, combined with ∇, yield considerably finer versions of the Shuffle conjecture.…”

“…To see how this comes about we briefly reproduce an argument first given in [16]. Exploiting (6.4) and (6.5), we calculate that…”

A new “dinv” arising from the two part case of the shuffle conjecture

“…More recently, J. Haglund, J. Morse and M. Zabrocki [12] formulated a variety of new conjectures yielding surprising refinements of the shuffle conjecture. In [12] they introduce a new ingredients in the Theory of parking functions.…”

“…, j + n in the main diagonal including car j + n in the cell (1, 1). In view of some recent conjectures of Haglund-Morse-Zabrocki [12] it is natural to conjecture that replacing E n,k by the modified Hall-Littlewood funtions Cp 1 Cp 2 · · · Cp k 1 would yield a polynomial that enumerates the same collection of parking functions but now restricted by the requirement that the Dyck path supporting cars j + 1, . .…”

“…In [12] they introduce a new ingredients in the Theory of parking functions. This is the diagonal composition of a Parking function, which we denote by p(P F ) and is simply the composition which gives the position of the zeros in the vector U = (u 1 , u 2 , .…”