Extending the Bipartite Parking Space

Abstract

We prove an analogue of a theorem of Berget and Rhoades about extending the parking space ${\mathrm{Park}}_n$ to an $S_{n+1}$-module ${\mathrm{Slim}}_n$. Specifically, we show that the \textit{bipartite parking space} ${\mathrm{Park}}_{K_{n,m}}$, which naturally comes with an $S_{n-1}\times S_m$ action, extends to an $S_n\times S_m$-representation ${\mathrm{Slim}}_{n,m}$. We then formulate a conjecture generalizing this statement to any simple graph.

13 pages, comments welcome!