Circular s-choice parking functions: an exact closed formula via rotational symmetry

Abstract

We study a circular variant of the $s$-choice parking model: $n$ cars park on $m=n+1$ spots arranged on a circle, each car carrying an anchor and $s-1$ clockwise increments at least $d$ apart with return gap at least $d$; a car tries its choices in order and then sweeps clockwise from its last choice. On the circle every car parks and exactly one spot remains empty. Exploiting rotational symmetry in the spirit of Pollak's proof of the count $(n+1)^{n-1}$, we prove that the empty spot is exactly equidistributed, which yields the closed formula $m^{n-1}\binom{m-sd+s-1}{s-1}^{n}$ for the number of preferences leaving any prescribed spot empty. This appears to be the first closed product formula in the multi-choice parking landscape. We further prove a refinement: within every class of preferences with prescribed increments, the empty spot is still exactly equidistributed, which explains the product structure of the formula and, for $s=2$, $d=1$, the appearance of the classical count $(n+1)^{n-1}$ as a factor of $(n+1)^{n-1}n^{n}$. The admissible tuples are enumerated through Kaplansky's lemma on circular selections, and all results are verified by exhaustive computer enumeration.