Abstract:

Parking functions are well-known objects in combinatorics.  There is an action of the symmetric group by permuting letters, which is quite important in algebraic combinatorics (it is related with Haiman's work on diagonal coinvariants).  The goal of this work is to study a poset of parking functions, introduced by Edelman in 1980, under the name of 2-noncrossing partitions. Edelman obtained nice formulas for the zeta polynomial and the enumeration by rank in the poset.  Generalizing the zeta polynomial, the character of the symmetric group acting on chains is also a nice object, related with the parking space theory of Armstrong, Reiner, and Rhoades. Our main result is that the poset is shellable, and from that we obtain the character of the symmetric group acting on the homology of the poset.

This is joint work with Bérénice Delcroix-Oger and Lucas Randazzo.