Generalized q,t-Catalan numbers
Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 855-886.

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: https://doi.org/10.5802/alco.120
Classification: 05A18,  05A19,  05E05
Keywords: q,t-Catalan numbers, symmetric chain decomposition, Khovanov–Rozansky knot homology.
@article{ALCO_2020__3_4_855_0,
     author = {Gorsky, Eugene and Hawkes, Graham and Schilling, Anne and Rainbolt, Julianne},
     title = {Generalized $q,t$-Catalan numbers},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {3},
     number = {4},
     year = {2020},
     pages = {855-886},
     doi = {10.5802/alco.120},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2020__3_4_855_0/}
}
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/item/ALCO_2020__3_4_855_0/

[1] Armstrong, Drew; Garsia, Adriano; Haglund, James; Rhoades, Brendon; Sagan, Bruce Combinatorics of Tesler matrices in the theory of parking functions and diagonal harmonics, J. Comb., Volume 3 (2012) no. 3, pp. 451-494 | Article | MR 3029443 | Zbl 1291.05203

[2] Ceballos, Cesar; González D’León, Rafael S. Signature Catalan combinatorics, J. Comb., Volume 10 (2019) no. 4, pp. 725-773 | Article | MR 3983746 | Zbl 1417.05003

[3] Elias, Ben; Hogancamp, Matthew On the computation of torus link homology, Compos. Math., Volume 155 (2019) no. 1, pp. 164-205 | Article | MR 3880028 | Zbl 07089331

[4] Garsia, A. M.; Haiman, M. A remarkable q,t-Catalan sequence and q-Lagrange inversion, J. Algebraic Combin., Volume 5 (1996) no. 3, pp. 191-244 | Article | MR 1394305 | Zbl 0853.05008

[5] Garsia, Adriano M.; Haglund, James; Xin, Guoce Constant term methods in the theory of Tesler matrices and Macdonald polynomial operators, Ann. Comb., Volume 18 (2014) no. 1, pp. 83-109 | Article | MR 3167606 | Zbl 1297.05240

[6] Gorsky, Eugene; Mazin, Mikhail Rational parking functions and LLT polynomials, J. Combin. Theory Ser. A, Volume 140 (2016), pp. 123-140 | Article | MR 3461138 | Zbl 1331.05227

[7] Gorsky, Eugene; Mazin, Mikhail; Vazirani, Monica Affine permutations and rational slope parking functions, Trans. Amer. Math. Soc., Volume 368 (2016) no. 12, pp. 8403-8445 | Article | MR 3551576 | Zbl 1346.05299

[8] Gorsky, Eugene; Mazin, Mikhail; Vazirani, Monica Rational Dyck paths in the non relatively prime case, Electron. J. Combin., Volume 24 (2017) no. 3, Paper 3.61, 29 pages | MR 3711103 | Zbl 1372.05017

[9] Gorsky, Eugene; Neguţ, Andrei Refined knot invariants and Hilbert schemes, J. Math. Pures Appl. (9), Volume 104 (2015) no. 3, pp. 403-435 | Article | MR 3383172 | Zbl 1349.14012

[10] Gorsky, Eugene; Neguţ, Andrei; Rasmussen, Jacob Flag Hilbert schemes, colored projectors and Khovanov–Rozansky homology (2016) (preprint https://arxiv.org/abs/1608.07308)

[11] Gorsky, Eugene; Oblomkov, Alexei; Rasmussen, Jacob; Shende, Vivek Torus knots and the rational DAHA, Duke Math. J., Volume 163 (2014) no. 14, pp. 2709-2794 | Article | MR 3273582 | Zbl 1318.57010

[12] Gorsky, Evgeny; Mazin, Mikhail Compactified Jacobians and q,t-Catalan numbers, I, J. Combin. Theory Ser. A, Volume 120 (2013) no. 1, pp. 49-63 | Article | MR 2971696 | Zbl 1252.05009

[13] Gorsky, Evgeny; Mazin, Mikhail Compactified Jacobians and q,t-Catalan numbers, II, J. Algebraic Combin., Volume 39 (2014) no. 1, pp. 153-186 | Article | MR 3144397 | Zbl 1284.05019

[14] Haglund, J.; Haiman, M.; Loehr, N.; Remmel, J. B.; Ulyanov, A. A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J., Volume 126 (2005) no. 2, pp. 195-232 | Article | MR 2115257 | Zbl 1069.05077

[15] Haglund, James The q,t-Catalan numbers and the space of diagonal harmonics, University Lecture Series, Volume 41, American Mathematical Society, Providence, RI, 2008, viii+167 pages (With an appendix on the combinatorics of Macdonald polynomials) | MR 2371044 | Zbl 1142.05074

[16] Haglund, James A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants, Adv. Math., Volume 227 (2011) no. 5, pp. 2092-2106 | Article | MR 2803796 | Zbl 1258.13020

[17] Hogancamp, Matthew Khovanov-Rozansky homology and higher Catalan sequences (2017) (preprint https://arxiv.org/abs/1704.01562)

[18] Kashiwara, Masaki The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J., Volume 71 (1993) no. 3, pp. 839-858 | Article | MR 1240605 | Zbl 0794.17008

[19] Lee, Kyungyong; Li, Li; Loehr, Nicholas A. Combinatorics of certain higher q,t-Catalan polynomials: chains, joint symmetry, and the Garsia-Haiman formula, J. Algebraic Combin., Volume 39 (2014) no. 4, pp. 749-781 | Article | MR 3199025 | Zbl 1293.05007

[20] Lee, Kyungyong; Li, Li; Loehr, Nicholas A. A combinatorial approach to the symmetry of q,t-Catalan numbers, SIAM J. Discrete Math., Volume 32 (2018) no. 1, pp. 191-232 | Article | MR 3747460 | Zbl 1386.05013

[21] Littelmann, Peter Crystal graphs and Young tableaux, J. Algebra, Volume 175 (1995) no. 1, pp. 65-87 | Article | MR 1338967 | Zbl 0831.17004

[22] Mellit, Anton Toric braids and (m,n)-parking functions (2016) (preprint https://arxiv.org/abs/1604.07456)

[23] Mellit, Anton Homology of torus knots (2017) (preprint https://arxiv.org/abs/1704.07630)

[24] Oblomkov, Alexei; Rozansky, Lev HOMFLYPT homology of Coxeter links (2017) (preprint https://arxiv.org/abs/1706.00124)

[25] Schiffmann, Olivier; Vasserot, Eric The elliptic Hall algebra and the K-theory of the Hilbert scheme of 𝔸 2 , Duke Math. J., Volume 162 (2013) no. 2, pp. 279-366 | Article | MR 3018956 | Zbl 1290.19001