Compactified Jacobians and q,t-Catalan numbers, II

We continue the study of the rational-slope generalized q,t-Catalan numbers cm,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 cm,n(q,1)=cm,n(1,q) for m=kn±1. We give a bijective proof of the full symmetry cm,n(q,t)=cm,n(t,q) for min(m,n)≤3. As a corollary of these combinatorial constructions, we give a simple formula for the Poincaré polynomials of compactified Jacobians of plane curve singularities xkn±1=yn. 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.

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