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Generalized q,t-Catalan numbers
- Source :
- Algebraic Combinatorics, vol 3, iss 4
- Publication Year :
- 2020
- Publisher :
- eScholarship, University of California, 2020.
-
Abstract
- Recent work of the first author, Negut and Rasmussen, and of Oblomkov and Rozansky in the context of Khovanov--Rozansky knot homology produces a family of polynomials in $q$ and $t$ labeled by integer sequences. These polynomials can be expressed as equivariant Euler characteristics of certain line bundles on flag Hilbert schemes. The $q,t$-Catalan numbers and their rational analogues are special cases of this construction. In this paper, we give a purely combinatorial treatment of these polynomials and show that in many cases they have nonnegative integer coefficients. For sequences of length at most 4, we prove that these coefficients enumerate subdiagrams in a certain fixed Young diagram and give an explicit symmetric chain decomposition of the set of such diagrams. This strengthens results of Lee, Li and Loehr for $(4,n)$ rational $q,t$-Catalan numbers.<br />Comment: 33 pages; v2: fixed typos and included referee comments
- Subjects :
- 0102 computer and information sciences
Homology (mathematics)
01 natural sciences
Combinatorics
Catalan number
symbols.namesake
FOS: Mathematics
05E05
Discrete Mathematics and Combinatorics
Mathematics - Combinatorics
0101 mathematics
math.CO
Mathematics::Symplectic Geometry
Mathematics
010102 general mathematics
Integer sequence
Mathematics::Geometric Topology
05A19
05A18
010201 computation theory & mathematics
Chain decomposition
Euler's formula
symbols
Equivariant map
Combinatorics (math.CO)
05A18, 05A19, 05E05
Knot (mathematics)
Subjects
Details
- Database :
- OpenAIRE
- Journal :
- Algebraic Combinatorics, vol 3, iss 4
- Accession number :
- edsair.doi.dedup.....70ee4cf6b75e7e1fac52df4a209df74e