Positroid Catalan numbers

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 $f$, we study the positroid Catalan number $C_f$ 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 $q,t$-polynomials coincide with the generalized $q,t$-Catalan numbers that recently appeared in relation to the shuffle conjecture, flag Hilbert schemes, and Khovanov-Rozansky homology of Coxeter links.