Positroid Catalan numbers
Pavel Galashin, Thomas Lam

TL;DR
This paper introduces a new class of permutations called repetition-free permutations and links their associated positroid Catalan numbers to counting specific Dyck paths, conjecturing a connection to generalized q,t-Catalan numbers.
Contribution
It defines repetition-free permutations and shows their positroid Catalan numbers count Dyck paths avoiding convex subsets, connecting combinatorics with algebraic geometry and knot homology.
Findings
Repetition-free permutations are introduced.
Positroid Catalan numbers count Dyck paths avoiding convex sets.
Conjectural link to generalized q,t-Catalan numbers.
Abstract
Given a permutation , we study the positroid Catalan number defined to be the torus-equivariant Euler characteristic of the associated open positroid variety. We introduce a class of repetition-free permutations and show that the corresponding positroid Catalan numbers count Dyck paths avoiding a convex subset of the rectangle. We show that any convex subset appears in this way. Conjecturally, the associated -polynomials coincide with the generalized -Catalan numbers that recently appeared in relation to the shuffle conjecture, flag Hilbert schemes, and Khovanov-Rozansky homology of Coxeter links.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Algebra and Geometry · Advanced Mathematical Identities
