Compactified Jacobians and q,t-Catalan numbers, II
By E. Gorsky, Mikhail Mazin. Journal of Algebraic Combinatorics, 2012
90 citations recorded by the source corpus.
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.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/7aa576a5a4bcff53ebc0d8a31f69d34770db2e6c
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
Citations in the corpus
- Refined knot invariants and Hilbert schemes (2013, background): “The affine Schubert cells in Grn cut out affine cells inΣ m n , which can be labeled either by the m–stable permutations with additional restrictions ([34]) or by the Dyck paths in t hem × n rectangle ([32, 33]).”
- DAHA and iterated torus knots (2014, background): “This formula is related to the perfect modules of rational DAHA; see eg [21].”
- Affine permutations and rational slope parking functions (2014, unknown): “In [15] the bijectivity of G was proved form = kn ± 1.”