Generalized $q,t$-Catalan numbers

Gorsky, Eugene; Hawkes, Graham; Schilling, Anne; Rainbolt, Julianne

doi:10.5802/alco.120

Generalized

q, t

-Catalan numbers

Gorsky, Eugene ¹; Hawkes, Graham ¹; Schilling, Anne ¹; Rainbolt, Julianne ²

¹ Department of Mathematics University of California One Shields Avenue Davis, CA 95616-8633, U.S.A.
² Department of Mathematics and Statistics Saint Louis University 220 North Grand Blvd. Saint Louis, MO 63103, U.S.A.

Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 855-886.

Abstract

Recent work of the first author, Neguţ and Rasmussen, and of Oblomkov and Rozansky in the context of Khovanov–Rozansky knot homology produces a family of polynomials in $q$ and $t$ labeled by integer sequences. These polynomials can be expressed as equivariant Euler characteristics of certain line bundles on flag Hilbert schemes. The $q, t$ -Catalan numbers and their rational analogues are special cases of this construction. In this paper, we give a purely combinatorial treatment of these polynomials and show that in many cases they have nonnegative integer coefficients.

For sequences of length at most 4, we prove that these coefficients enumerate subdiagrams in a certain fixed Young diagram and give an explicit symmetric chain decomposition of the set of such diagrams. This strengthens results of Lee, Li and Loehr for $(4, n)$ rational $q, t$ -Catalan numbers.

Received: 2019-06-01
Revised: 2020-02-07
Accepted: 2020-03-19
Published online: 2020-08-20

DOI: 10.5802/alco.120

Classification: 05A18, 05A19, 05E05
Keywords: $q,t$-Catalan numbers, symmetric chain decomposition, Khovanov–Rozansky knot homology.

Author's affiliations:

Gorsky, Eugene ¹; Hawkes, Graham ¹; Schilling, Anne ¹; Rainbolt, Julianne ²

¹ Department of Mathematics University of California One Shields Avenue Davis, CA 95616-8633, U.S.A.
² Department of Mathematics and Statistics Saint Louis University 220 North Grand Blvd. Saint Louis, MO 63103, U.S.A.

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Generalized $q,t${-Catalan} numbers},
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Gorsky, Eugene; Hawkes, Graham; Schilling, Anne; Rainbolt, Julianne. Generalized $q,t$-Catalan numbers. Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 855-886. doi : 10.5802/alco.120. https://alco.centre-mersenne.org/articles/10.5802/alco.120/

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