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

Abstract: Unit-interval parking functions of length $n$ are enumerated by the Fubini numbers $F_n$ and are in explicit bijection with the ordered set partitions of $[n]$. We use this bijection to single out the unit-interval parking functions whose associated ordered set partition is \emph{deranged} in the sense of Belbachir, Djemmada, and Németh -- no block occupies the position indexed by its minimum element -- and call them the \emph{deranged unit-interval parking functions} $\DUPF_n$. Since the bijection restricts to the deranged objects, $|\DUPF_n| = \tilde F_n$, the $n$-th deranged Bell number. We give an intrinsic, coordinate-wise characterization of the deranged condition through the lucky cars (equivalently, the block leaders) of a parking function, and we refine the enumeration by total displacement, obtaining $d_m\Stir{n}{m}$ deranged unit-interval parking functions with $m$ blocks. We derive the exponential generating function $e^{1-e^x}/(2-e^x)$ by the symbolic method, prove a fixed-block convolution relating $F_n$, the Bell numbers, and $\tilde F_n$, refine the count by singleton blocks via $2$-associated Stirling numbers, and describe an $r$-start extension together with a deranged Cayley-permutation model. Worked examples and a table of values are included.