The Combinatorics of Affine Deodhar Diagrams

Deodhar diagrams give a combinatorial way to compute point counts of open positroid varieties over finite fields. We introduce affine Deodhar diagrams, which extend this construction to affine patches of open positroid varieties. These diagrams are indexed by a bounded affine permutation together with a lattice path, and this extra flexibility makes them especially useful for recursive bijections. Motivated by connections with Dyck paths, open positroid varieties, and their cluster structure, we construct bijections between several classes of affine Deograms. These bijections give combinatorial proofs of point-count identities that were previously known from geometric isomorphisms.