Compactified Jacobians and q,t-Catalan numbers, II

TL;DR

This paper advances the understanding of rational-slope generalized q,t-Catalan numbers by establishing symmetry properties, providing combinatorial proofs, and connecting these numbers to geometric invariants of plane curve singularities.

Contribution

It generalizes bijective constructions to prove symmetry properties of rational-slope q,t-Catalan numbers and links them to the geometry of compactified Jacobians.

Findings

Proved weak symmetry c_{m,n}(q,1)=c_{m,n}(1,q) for m=kn±1

Established full symmetry c_{m,n}(q,t)=c_{m,n}(t,q) for min(m,n)≤3

Derived a simple formula for Poincaré polynomials of compactified Jacobians

Abstract

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 $\min(m,n)\le 3$. As a corollary of these combinatorial constructions, we give a simple formula for the Poincar\'e 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.