Extending the parking space

2013/03/22 by Berget, Andrew, Rhoades, Brendon · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1303.5505

Abstract

The action of the symmetric group S_n on the set Park_n of parking functions of size n has received a great deal of attention in algebraic combinatorics. We prove that the action of S_n on Park_n extends to an action of S_n+1. More precisely, we construct a graded S_n+1-module V_n such that the restriction of V_n to S_n is isomorphic to Park_n. We describe the S_n-Frobenius characters of the module V_n in all degrees and describe the S_n+1-Frobenius characters of V_n in extreme degrees. We give a bivariate generalization V_n^{(ℓ, m)} of our module V_n whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization.

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• Short survey of results and open problems for parking problems on random trees

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