On Deranged Unit-Interval Parking Functions and the Deranged Bell Numbers

Abstract

Unit-interval parking functions are counted by the Fubini numbers and are in explicit bijection with ordered set partitions. We transport the deranged ordered set partitions of Belbachir, Djemmada, and Németh through this bijection and obtain the deranged unit-interval parking functions $\mathrm{DUPF}_n$. The equality $|\mathrm{DUPF}_n|=\widetilde F_n$, the Stirling-transform formula, the exponential generating function $e^{1-e^x}/(2-e^x)$, and the dominant asymptotics are therefore not presented as new enumerative discoveries; they are consequences of the known deranged Bell-number theory. The new material of this note is the parking-side structure: leader and lucky-car characterizations, a fixed-block stratification of all unit-interval parking functions, rencontres-type generating functions and a Poisson limit law for fixed blocks, a bijective fixed-block decomposition of the Fubini numbers, a multivariate block-size refinement, a fully deranged $r$-start extension, and a Cayley-permutation model based on first appearances.