Conjectured statistics for the q,t-Catalan numbers

By James Haglund. 2003

120 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/da43e7c06aca7535ebd603039acf0d006cd8d072

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Citations in the corpus

A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path (2010, background): “ of [HHLRU05] which asserts that the coecient of m in re nsimplies to a q;tstatistic on lattice paths. A major breakthrough in this direction was made with the conjectured combinatorial formula of [Hag03] for the coecient of m 1n = s 1n in re n. In this case, the coecient is a bi-graded version of t”
Compactified Jacobians and q,t-Catalan numbers, II (2012, unknown): “The 5–generators of ∆ are{a0, a1, a2, a3, a4} = {0, 7, 8, 11, 14}, and g5(0) = ♯([0, 7) \∆) = 5, g5(7) = ♯([7, 14) \∆) = 1, g5(8) = ♯([8, 15) \∆) = 1, andg5(11) = g5(14) = 0.”
The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link (2012, result): “BINATORICS OF HOMFLY HOMOLOGY, BY EUGENE GORSKY This appendix relates the combinatorics of the cells in the Hilbert scheme of a plane curve singularity with one Puiseaux pair to the existing results ([Hag03],[Hag04],[EHKK03]) on the combinatorics of diagonal harmonics and DAHA representations. This ”
A combinatorial formula for the character of the diagonal coinvariants (2003, methodology): “ that the right-hand side of (47) decomposes as (49) X λ (t;t)nsλ(1/(1− t))sλ(z), which is equal to ∇en(z)|q=0 by Lemma 4.2.1 and the Cauchy formula. 4.4. The q,t-Catalan formula. Haglund conjectured [7] and with Garsia proved [3, 4] a formula for the q,t-Catalan polynomial (50) Cn(q,t) = h∇en,eni. ”
The representation of the symmetric group on m-Tamari intervals (2012, unknown): “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].”
A proof of the q,t-Schröder conjecture (2004, background): “The statistic bounce was first introduced in [8] (although called differently there).”