A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

By James Haglund, Jennifer Morse, Mike Zabrocki. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2010

82 citations recorded by the source corpus.

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 $\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. 126 (2005), pp. 195 − 232) for $\nabla {{e}_{n}}\left[ X \right]$ . 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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7dc589087d5059bc4f4ca61205766dc8d0931ecd

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Citations in the corpus

A Lift of the Schur and Hall–Littlewood Bases to Non-commutative Symmetric Functions (2012, methodology): “These functions were recently studied further in [15] where it was shown that”
The representation of the symmetric group on m-Tamari intervals (2012, background): “They seem to parallel the (now largely proved) connections between ballot paths and bivariate diagonal coinvariant spaces, which have attracted considerable attention in the past 20 years [14, 17, 18, 21, 24, 23, 29] and are still a very active area of research today [1, 2, 13, 16, 22, 19, 25, 28].”
The Delta Conjecture (2015, background): “(108) where Cα is a certain symmetric function with coefficients in Q(q) which is defined in [HMZ12].”