Extending the parking space
Andrew Berget, Brendon Rhoades

TL;DR
This paper extends the symmetric group action on parking functions to a larger group, constructing a new module with detailed character descriptions and generalizations involving bivariate and Fuss parameters.
Contribution
It constructs an $S_{n+1}$-module extending the $S_n$-action on parking functions and provides explicit Frobenius character formulas, including bivariate and Fuss generalizations.
Findings
Constructed a graded $S_{n+1}$-module $V_n$ extending the parking function action.
Described Frobenius characters of $V_n$ in all degrees.
Introduced a bivariate generalization $V_n^{(\, ext{ell, m)}}$ with connections to Dyck paths.
Abstract
The action of the symmetric group on the set of parking functions of size has received a great deal of attention in algebraic combinatorics. We prove that the action of on extends to an action of . More precisely, we construct a graded -module such that the restriction of to is isomorphic to . We describe the -Frobenius characters of the module in all degrees and describe the -Frobenius characters of in extreme degrees. We give a bivariate generalization of our module 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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Taxonomy
TopicsAdvanced Combinatorial Mathematics · graph theory and CDMA systems · Coding theory and cryptography
