Some natural extensions of the parking space

We construct a family of Sn modules indexed by c?{1,…,n} with the property that upon restriction to Sn?1 they recover classical parking function representation Haiman. The construction these relies on an Sn-action set is closely related functions. compute characters and use resulting description classify them up isomorphism. In particular, we show number isomorphism classes equal divisors d n satisfying d?2(mod4). cases c=n c=1, orbits. Based empirical evidence, conjecture when our h-positive in fact (ungraded) extension constructed Berget Rhoades.