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

Summary · qwen2.5:32b

The article establishes a bijection between deranged ordered set partitions and deranged unit-interval parking functions ($\mathrm{DUPF}_n$), showing that their count is given by the deranged Bell numbers $\widetilde F_n$. This work introduces new structural insights into parking functions, including leader and lucky-car characterizations, a fixed-block stratification, and generating functions related to these structures.

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Exploring the world of deranged unit-interval parking functions in mathematics.

Excerpt

arXiv:2607.01273v2 Announce Type: replace 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 N\'emeth 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.