The Catalan numbers have many interpretations (counting Dyck paths, triangulations, etc). One particularly well-known refinement comes from the Narayana numbers, which can be interpreted as counting Dyck paths with a fixed number of peaks. In this talk we introduce a geometric interpretation of the Narayana numbers, via the characteristic polynomial of the dominant cone of the Shi arrangement, and use the geometry to give a geometric refinement of the number of parking functions.
Narayana numbers from the geometry of Shi arrangements*
Galen Dorpalen-Barry, Ruhr-University BochumAuthors: Galen Dorpalen-Barry, Christian Stump
2022 AWM Research Symposium
Women from the Graduate Research Workshop in Combinatorics