A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

http://dx.doi.org/10.4153/CJM-2011-078-4
Canad. J. Math. 64(2012), 822-844

  Published:2011-10-22
 Printed: Aug 2012

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 $\nabla$ 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. $\mathbf {126}$ ($2005$), 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.