Geodesic
Compactified Jacobians and $q,t$-Catalan numbers. II
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 1, pp. 153-186.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

We continue the study of the rational-slope generalized $q,t$-Catalan numbers $c _{m,n}(q,t)$. We describe generalizations of the bijective constructions of J. Haglund and N. Loehr and use them to prove a weak symmetry property $c _{m,n}(q,1)=c _{m,n}(1,q)$ for $m=kn\pm 1$. We give a bijective proof of the full symmetry $c _{m,n}(q,t)=c _{m,n }(t,q)$ for $\min(m,n)\leq 3$. As a corollary of these combinatorial constructions, we give a simple formula for the Poincaré polynomials of compactified Jacobians of plane curve singularities $x ^{kn\pm 1}=y ^{n}$. We also give a geometric interpretation of a relation between rational-slope Catalan numbers and the theory of $(m,n)$-cores discovered by J. Anderson. For Part I see [J. Comb. Theory, Ser. A 120, No. 1, 49--63 (2013; Zbl 1252.05009)].
Classification : 05A15
Keywords: $q,t$-Catalan numbers, compactified Jacobian, semigroup 
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     author = {Gorsky, Evgeny and Mazin, Mikhail},
     title = {Compactified {Jacobians} and $q,t${-Catalan} numbers. {II}},
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Gorsky, Evgeny; Mazin, Mikhail. Compactified Jacobians and $q,t$-Catalan numbers. II. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 1, pp. 153-186. http://geodesic.mathdoc.fr/item/JAC_2014__39_1_a2/