Preference-Restricted Parking Functions

• Preference-restricted parking functions are a refined class where the allowable parking preferences are constrained, preserving the Catalan condition while enabling novel enumerative formulations.
• They provide a unified framework connecting classical, prime, and defective parking functions with models like unit interval, Naples, and vacillating parking functions through systematic preference restrictions.
• The approach yields combinatorial proofs of classical identities, such as Abel’s binomial theorem, and links to ordered set partitions, Fubini numbers, and the face structure of permutohedra.

Preference-restricted parking functions are refinements of classical parking functions in which admissible preferences are constrained more tightly than in the ordinary model. In one formulation, an $S$-restricted parking function on $n$ cars is a parking function $\pi:[n]\to S\subseteq[n]$; in another, the restriction is imposed by a local parking rule, as in unit interval parking functions, where each car may park only in its preferred spot or the immediately following spot (Bown et al., 15 Jul 2025, Bradt et al., 2024). Across these formulations, the subject connects Catalan-type inequalities, prime decompositions, ordered set partitions, Fubini numbers, $\mathbf u$-parking functions, and permutohedra.

1. Classical criterion and codomain restriction

A classical parking function on $n$ cars is a function

$\pi:[n]\to[n]$

whose $i$th-smallest output is at most $i$. Equivalently, if $\pi^\uparrow$ is the nondecreasing rearrangement of $\pi$, then

$n$0

This is the standard sorted-form characterization, referred to in the restricted-setting paper as the Catalan condition; the classical count is

$n$1

An $n$2-restricted parking function is then simply a parking function whose image is contained in a prescribed subset $n$3, and the corresponding set is denoted $n$4 (Bown et al., 15 Jul 2025).

This codomain-restriction viewpoint isolates the role of the allowable preference set without changing the underlying parking procedure. The ordinary case is recovered by taking $n$5. Because the Catalan condition remains in force, the restriction acts by intersecting the usual parking-function class with a coordinatewise image constraint. The 2025 formulation emphasizes that many apparently different parking variants can be reformulated in this way, so that preference restriction becomes a common combinatorial language rather than a single isolated subclass (Bown et al., 15 Jul 2025).

2. Prime restrictions and defective parking

A parking function $n$6 is prime if for every $n$7,

$n$8

Equivalently, in sorted form,

$n$9

The restricted theory defines $\pi:[n]\to S\subseteq[n]$0 analogously: these are prime parking functions with image contained in $\pi:[n]\to S\subseteq[n]$1 (Bown et al., 15 Jul 2025).

A central structural theorem identifies prime parking functions with an ordinary restricted class: $\pi:[n]\to S\subseteq[n]$2 The bijection is given by shifting every preference $\pi:[n]\to S\subseteq[n]$3 up by $\pi:[n]\to S\subseteq[n]$4,

$\pi:[n]\to S\subseteq[n]$5

and it extends more generally as follows. If $\pi:[n]\to S\subseteq[n]$6, then $\pi:[n]\to S\subseteq[n]$7-restricted prime parking functions are in bijection with $\pi:[n]\to S\subseteq[n]$8-restricted parking functions, where

$\pi:[n]\to S\subseteq[n]$9

This converts strict Catalan conditions into ordinary Catalan conditions after a systematic preference shift (Bown et al., 15 Jul 2025).

Initial-segment restrictions provide a second major specialization. For

$\mathbf u$0

the model is linked to parking with fewer spots than cars. If there are $\mathbf u$1 cars and only $\mathbf u$2 spots, then the minimum possible defect is $\mathbf u$3, and preference functions $\mathbf u$4 with minimum possible defect $\mathbf u$5 are in bijection with $\mathbf u$6-restricted parking functions. This gives a direct reformulation of defective parking in restricted-preference terms (Bown et al., 15 Jul 2025).

3. Enumerative formulas and Abel-type identities

For $\mathbf u$7-restricted parking functions, two complementary counting formulas are given. The first is obtained by counting all maps $\mathbf u$8 and subtracting the non-parking ones: $\mathbf u$9 The second arises from a sign-reversing involution on 2-colored parking functions: $n$0 The prime analogue is similarly doubled: $n$1 and also

$n$2

These identities organize restricted parking counts by the first failure of the Catalan or strict Catalan condition (Bown et al., 15 Jul 2025).

Equating the two formulas for $n$3 yields a combinatorial proof of Abel’s binomial theorem in the form

$n$4

A specialization obtained in the paper is

$n$5

and the prime formulas similarly imply

$n$6

The restricted model therefore functions არა only as an enumeration problem but also as a source of Abel-type identities (Bown et al., 15 Jul 2025).

The same paper refines these counts by the number of cars preferring spot $n$7. For ordinary parking functions,

$n$8

while for $n$9-restricted parking functions one has

$\pi:[n]\to[n]$0

equivalently

$\pi:[n]\to[n]$1

These refinements recover Abel’s theorem by polynomial interpolation in $\pi:[n]\to[n]$2 (Bown et al., 15 Jul 2025).

4. Symmetry, outcomes, and $\pi:[n]\to[n]$3-parking interpretations

The restricted model is compatible with several further combinatorial structures. Under the natural action of $\pi:[n]\to[n]$4 by permuting cars, the number of $\pi:[n]\to[n]$5-orbits in $\pi:[n]\to[n]$6 is

$\pi:[n]\to[n]$7

the $\pi:[n]\to[n]$8-entry of Catalan’s triangle, and the orbit representatives are precisely the nondecreasing $\pi:[n]\to[n]$9-restricted parking functions. The proof uses the recurrence

$i$0

The same paper also refines counts by parking outcome: for $i$1, the number of $i$2-restricted parking functions with outcome $i$3 is

$i$4

where $i$5 is the length of the longest contiguous block ending at position $i$6 whose maximum is $i$7 (Bown et al., 15 Jul 2025).

Preference restriction also models multiple-capacity parking. If spots are grouped into rows of size $i$8, and cars may prefer only the first spot of each row,

$i$9

then for $i$0 spots the number of such restricted parking functions is

$i$1

More generally, for $i$2, the paper gives a recursive decomposition for $i$3 cars/spots: $i$4 This recovers the Blake–Konheim enumeration in restricted-preference form (Bown et al., 15 Jul 2025).

A further equivalence identifies restricted parking functions with a standard generalized model. If $i$5 and

$i$6

then $i$7-restricted parking functions are in bijection with the corresponding $i$8-parking functions (Bown et al., 15 Jul 2025). Subsequent work on vector parking functions refines this by lucky cars: for a fixed lucky set $i$9 and fixed outcome $\pi^\uparrow$0, the number of possible $\pi^\uparrow$1-parking functions is

$\pi^\uparrow$2

so that

$\pi^\uparrow$3

The same line of work counts outcomes by lucky spots through multinomial coefficients and gives blockwise formulas for $\pi^\uparrow$4-parking functions with exactly $\pi^\uparrow$5 lucky cars (Ferreri et al., 19 Aug 2025).

5. Unit interval parking functions and Fubini phenomena

Unit interval parking functions form the most rigid preference-restricted subclass treated explicitly as such in the recent literature. A unit interval parking function of length $\pi^\uparrow$6 is a parking function in which each car parks either in its preferred spot or in the spot immediately after it. The set is denoted $\pi^\uparrow$7. Thus the parking rule allows only displacement $\pi^\uparrow$8 or $\pi^\uparrow$9; for example,

$\pi$0

This family is naturally equivalent to Fubini rankings, and the bijection yields

$\pi$1

where $\pi$2 is the $\pi$3th Fubini number. The same work gives the formulas

$\pi$4

and

$\pi$5

and proves that a rearrangement of a unit interval parking function remains unit interval if and only if it preserves the relative order within each block of the weakly increasing rearrangement (Bradt et al., 2024).

A more geometric analysis introduces displacement

$\pi$6

and shows that for unit interval parking functions each $\pi$7. In parking order, the prime objects are completely rigid: $\pi$8 is prime if and only if

$\pi$9

Hence there is exactly one prime unit-interval parking function of each length, and any parking-ordered unit-interval parking function decomposes uniquely into prime blocks of the form

$n$00

If the prime decomposition has block sizes $n$01, then

$n$02

This yields

$n$03

so unit interval parking functions with displacement $n$04 are counted by ordered set partitions of $n$05 into $n$06 blocks (Meyles et al., 2023).

The same displacement statistic is identified with permutohedral face dimension. For all integers $n$07, the unit-interval parking functions of length $n$08 with total displacement $n$09 are in bijection with the $n$10-dimensional faces of the permutohedron of order $n$11. If the prime decomposition has block sizes $n$12, the corresponding face has combinatorial type

$n$13

This gives an explicit bridge among unit interval parking functions, ordered set partitions, Fubini numbers, and the face structure of the permutohedron (Meyles et al., 2023).

6. Other local-rule restrictions: Naples and vacillation

The preference-restricted viewpoint also extends to parking models defined by nonclassical motion rules. In the $n$14-Naples setting, the 2023 paper "On the combinatorics of $n$15-Naples parking functions and parking strategies" characterizes $n$16-Naples parking functions in terms of subsequences with the structure of a complete $n$17-Naples parking function, defines complete parking preferences by requiring that for all $n$18, the number of cars having preference at least $n$19 is strictly greater than the number of spots in $n$20, and gives a characterization of permutation invariant $n$21-Naples parking functions. It also introduces parking strategies as vectors of rules that allow all cars to park and studies strategies minimizing natural parameters such as the total number of backward steps or the number of cars that need to drive backwards (Verciani, 2023).

Vacillating parking functions impose a different local constraint. For fixed integers $n$22, a car with preference $n$23 checks the spots

$n$24

in that order, parking in the first available one if it exists. The corresponding set is denoted $n$25. If

$n$26

then the main enumeration theorem gives

$n$27

For nondecreasing vacillating parking functions,

$n$28

and this sequence equals the numerator of the $n$29th convergent of $n$30, with closed form

$n$31

These results place preference restriction in a broader landscape of rigid local parking rules whose combinatorics is governed by decompositions, recurrence relations, and classical integer sequences (Fang et al., 2024).

Taken together, these developments show that preference restriction is not a single construction but a family of closely related mechanisms. In the strict codomain form, it turns parking-function theory into a flexible framework for prime objects, defective parking, multiple-capacity models, and Abel-type identities. In the local-rule form, it produces rigid subclasses such as unit interval, Naples, and vacillating parking functions, where displacement, block structure, and parking strategies become the primary invariants.