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In our recent preprint with Eugene Gorsky and Alexey Oblomkov we constructed affine cell decompositions of the Compactified Jacobians of plane curve singularities admitting a parametrization of the form x=t^{dn}, y= t^{dm}+at^{dm+1}+..., where n and m are relatively prime, d,n,m>1, and a is not zero. The cells turned out to be enumerated by (dn,dm)-invariant subsets of the non-negative integers, satisfying a certain admissibility condition. Furthermore, we showed that these admissible subsets are in bijection with the rational (dn,dm)-Dyck paths, i.e. South-East lattice paths that stay under the diagonal of an dn x dm rectangle, and that under this bijection the dimension of a cell is equal to the codinv of the corresponding Dyck path. This part of the work depends on an earlier paper joint with Eugene Gorsky and Monica Vazirani, where we construct an equivalence relation on the set of (dn,dm)-invariant subsets.

In this talk I will focus on the combinatorial part of our paper with Eugene and Alexey. I will introduce the equivalence relation on the invariant subsets, show that each equivalence class contains a unique admissible representative, and explain a bijection between the equivalence classes and the Dyck paths.