Andrew Berget ; Brendon Rhoades

Extending the parking space
dmtcs:2325 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2013,
DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)

https://doi.org/10.46298/dmtcs.2325
Extending the parking space
Authors: Andrew Berget ^{1}; Brendon Rhoades ^{2}
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Andrew Berget;Brendon Rhoades
1 Department of Mathematics [Seattle]
2 Department of Mathematics [Univ California San Diego]
The action of the symmetric group $S_n$ on the set $\mathrm{Park}_n$ 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 $\mathrm{Park}_n$ 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 $\mathrm{Park}_n$. 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 $V_n^{(\ell, m)}$ 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.