Geodesic
A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path
Canadian journal of mathematics, Tome 64 (2012) no. 4, pp. 822-844

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DOI

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.
DOI : 10.4153/CJM-2011-078-4
Mots-clés : 05E05, 33D52, Dyck Paths, Parking functions, Hall–Littlewood symmetric functions 
Haglund, J.; Morse, J.; Zabrocki, M. A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path. Canadian journal of mathematics, Tome 64 (2012) no. 4, pp. 822-844. doi: 10.4153/CJM-2011-078-4
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     journal = {Canadian journal of mathematics},
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