Compactified Jacobians and q,t-Catalan numbers, II

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±1. We give a bijective proof of the full symmetry c _m,n (q,t)=c _m,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 x ^kn±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.