Vector Parking Functions

• Vector parking functions are generalizations of classical parking functions defined by a vector of thresholds that determine parking spot capacities.
• They are characterized by diverse models—including lattice paths, tree bijections, and decomposition techniques—that yield rich enumerative formulas and structural insights.
• The theory connects with broader frameworks such as graphical, matrix, and hypergraph parking functions, and extends to higher-dimensional and restricted settings.

Searching arXiv for recent and foundational papers on vector parking functions and closely related generalizations. Vector parking functions are a family of generalizations of classical parking functions in which the admissibility condition is controlled by a vector of thresholds rather than by the classical diagonal bound. In one standard formulation, a weakly increasing vector $u$ prescribes a right boundary for lattice paths and, simultaneously, the capacities of parking spots; in another, a vector $x$ prescribes cumulative bounds on order statistics. These objects retain the core parking-process interpretation of the classical theory while supporting richer enumerative formulas, prime decompositions, tree and forest bijections, asymptotic regimes, and higher-dimensional extensions (Armon et al., 2024, Gaydarov et al., 2015).

1. Definitions and conventions

Two closely related notational systems are standard in the literature. In the weakly increasing boundary-vector convention, one fixes

$$u=(u_0,u_1,\dots,u_{n-1}), \qquad 1\le u_0\le u_1\le \cdots \le u_{n-1},$$

and calls a sequence

$a=(a_0,a_1,\dots,a_{n-1})\in \mathbb{N}^n$

a $u$-parking function if its order statistics satisfy

$a_{(i)}<u_i \qquad \text{for } i=0,1,\dots,n-1.$

Equivalently,

$\#\{j: a_j < u_i\}\ge i+1,\qquad i=0,\dots,n-1.$

The corresponding sets are denoted $PF(u)$ and $IPF(u)$ for all and increasing $u$-parking functions, respectively (Armon et al., 2024).

In the $x$0-parking-function convention, one fixes

$x$1

and requires the nondecreasing rearrangement

$x$2

to satisfy

$x$3

The literature also uses the term “vector parking functions” for these $x$4-dependent objects, and one paper explicitly notes that what had been called generalized parking functions are better called vector parking functions because they depend on a choice of vector $x$5 (Yang, 2024, Gaydarov et al., 2015).

Classical parking functions arise as special cases of both conventions. In the boundary-vector notation, they correspond to

$x$6

so the condition becomes $x$7. In the $x$8-notation, they correspond to

$x$9

so the cumulative bound is again $$u=(u_0,u_1,\dots,u_{n-1}), \qquad 1\le u_0\le u_1\le \cdots \le u_{n-1},$$0 after the indexing shift is accounted for (Armon et al., 2024, Gaydarov et al., 2015).

The conventions are not completely uniform across papers. Some treatments assume a strictly increasing vector

$$u=(u_0,u_1,\dots,u_{n-1}), \qquad 1\le u_0\le u_1\le \cdots \le u_{n-1},$$1

while others allow weakly increasing boundary data, precisely because repeated values encode parking-spot capacities (Yin, 2021, Armon et al., 2024). This difference is substantive rather than cosmetic: the weakly increasing model supports the direct capacity interpretation, whereas the strictly increasing model is especially convenient for multi-shuffle and asymptotic analysis.

2. Parking processes, lattice paths, and tree models

The parking-process interpretation is explicit in the weakly increasing model. The vector $$u=(u_0,u_1,\dots,u_{n-1}), \qquad 1\le u_0\le u_1\le \cdots \le u_{n-1},$$2 encodes spot capacities: the capacity of spot $$u=(u_0,u_1,\dots,u_{n-1}), \qquad 1\le u_0\le u_1\le \cdots \le u_{n-1},$$3 is the multiplicity of $$u=(u_0,u_1,\dots,u_{n-1}), \qquad 1\le u_0\le u_1\le \cdots \le u_{n-1},$$4 in $$u=(u_0,u_1,\dots,u_{n-1}), \qquad 1\le u_0\le u_1\le \cdots \le u_{n-1},$$5, denoted $$u=(u_0,u_1,\dots,u_{n-1}), \qquad 1\le u_0\le u_1\le \cdots \le u_{n-1},$$6. Cars arrive with preferences $$u=(u_0,u_1,\dots,u_{n-1}), \qquad 1\le u_0\le u_1\le \cdots \le u_{n-1},$$7, and car $$u=(u_0,u_1,\dots,u_{n-1}), \qquad 1\le u_0\le u_1\le \cdots \le u_{n-1},$$8 parks at its preferred spot or the next available spot with remaining capacity. In the equivalent 1-based notation, the capacity of spot $$u=(u_0,u_1,\dots,u_{n-1}), \qquad 1\le u_0\le u_1\le \cdots \le u_{n-1},$$9 is

$a=(a_0,a_1,\dots,a_{n-1})\in \mathbb{N}^n$0

and unavailable spots are precisely those with zero multiplicity (Armon et al., 2024, Ferreri et al., 19 Aug 2025).

A lattice-path model gives a geometric encoding. For a weakly increasing vector $a=(a_0,a_1,\dots,a_{n-1})\in \mathbb{N}^n$1, let $a=(a_0,a_1,\dots,a_{n-1})\in \mathbb{N}^n$2 be the unique path whose vertical steps have $a=(a_0,a_1,\dots,a_{n-1})\in \mathbb{N}^n$3-coordinates

$a=(a_0,a_1,\dots,a_{n-1})\in \mathbb{N}^n$4

Then a sequence $a=(a_0,a_1,\dots,a_{n-1})\in \mathbb{N}^n$5 is a $a=(a_0,a_1,\dots,a_{n-1})\in \mathbb{N}^n$6-parking function if and only if the lattice path $a=(a_0,a_1,\dots,a_{n-1})\in \mathbb{N}^n$7 lies to the left of $a=(a_0,a_1,\dots,a_{n-1})\in \mathbb{N}^n$8, i.e. has right boundary $a=(a_0,a_1,\dots,a_{n-1})\in \mathbb{N}^n$9. This turns the parking inequalities into a boundary-avoidance condition and later becomes the basis for prime decomposition (Armon et al., 2024).

Several non-path models are also central. For $u$0, Yan’s bijection identifies $u$1-parking functions with rooted labeled forests whose edges carry colors, with a distinguished root $u$2, with children of the root having $u$3 color choices and all other edges having $u$4 color choices. In the same paper, the statistic $u$5, the number of entries equal to $u$6, is interpreted through the root structure of the associated forest (Yang, 2024).

A different structural encoding is given by the depth-first-search variant of Dhar’s burning algorithm. For $u$7-parking functions, this produces rooted plane trees together with admissible vertex orders, and yields the identity

$u$8

where $u$9. This places vector parking functions in direct correspondence with weighted inversion statistics on rooted plane trees (Gaydarov et al., 2015).

These encodings show that vector parking functions are not merely a reformulation of a parking procedure. They admit boundary, forest, and tree realizations, each isolating a different structural feature: capacities, decomposition, inversion statistics, or root-based refinements.

3. Enumeration and explicit formulas

The most classical explicit formulas occur when the boundary vector is an arithmetic progression,

$a_{(i)}<u_i \qquad \text{for } i=0,1,\dots,n-1.$0

In that case,

$a_{(i)}<u_i \qquad \text{for } i=0,1,\dots,n-1.$1

When $a_{(i)}<u_i \qquad \text{for } i=0,1,\dots,n-1.$2, these specialize to the classical counts for all parking functions and increasing parking functions (Armon et al., 2024).

The same total count appears in the $a_{(i)}<u_i \qquad \text{for } i=0,1,\dots,n-1.$3-model for

$a_{(i)}<u_i \qquad \text{for } i=0,1,\dots,n-1.$4

where it is recalled as the Pitman–Stanley formula: $a_{(i)}<u_i \qquad \text{for } i=0,1,\dots,n-1.$5 In the forest model, this count is explained by the $a_{(i)}<u_i \qquad \text{for } i=0,1,\dots,n-1.$6 color choices on edges adjacent to the root and the $a_{(i)}<u_i \qquad \text{for } i=0,1,\dots,n-1.$7 color choices on all other edges (Yang, 2024).

For general strictly increasing $a_{(i)}<u_i \qquad \text{for } i=0,1,\dots,n-1.$8, the count is expressed through Gončarov polynomials: $a_{(i)}<u_i \qquad \text{for } i=0,1,\dots,n-1.$9 The same paper also gives a composition formula

$\#\{j: a_j < u_i\}\ge i+1,\qquad i=0,\dots,n-1.$0

where $\#\{j: a_j < u_i\}\ge i+1,\qquad i=0,\dots,n-1.$1 is the set of compositions $\#\{j: a_j < u_i\}\ge i+1,\qquad i=0,\dots,n-1.$2 satisfying

$\#\{j: a_j < u_i\}\ge i+1,\qquad i=0,\dots,n-1.$3

This formula is derived by decomposing parking functions according to empty parking spots and reducing the problem to products of classical counts on independent segments (Yin, 2021).

The same source introduces the parking function multi-shuffle. Fixing $\#\{j: a_j < u_i\}\ge i+1,\qquad i=0,\dots,n-1.$4, if the maximal admissible initial coordinates are

$\#\{j: a_j < u_i\}\ge i+1,\qquad i=0,\dots,n-1.$5

then the suffix $\#\{j: a_j < u_i\}\ge i+1,\qquad i=0,\dots,n-1.$6 is characterized as a shuffle of smaller parking functions on shifted subintervals. The resulting theorem identifies exactly when a partially specified word extends to a $\#\{j: a_j < u_i\}\ge i+1,\qquad i=0,\dots,n-1.$7-parking function, and it yields explicit formulas for the number of parking functions with prescribed initial coordinates (Yin, 2021).

There are also determinant formulas in special families. In the hypergraph setting, complete hypergraphs $\#\{j: a_j < u_i\}\ge i+1,\qquad i=0,\dots,n-1.$8 give rise to explicit vectors

$\#\{j: a_j < u_i\}\ge i+1,\qquad i=0,\dots,n-1.$9

and

$PF(u)$0

for which the $PF(u)$1-parking functions are exactly the $PF(u)$2-parking functions in the sense of Yan. The paper recalls the Stanley–Pitman/Steck determinant formula

$PF(u)$3

for these counts (Blanton et al., 13 Aug 2025).

4. Prime vector parking functions and canonical decomposition

Prime parking functions are the indecomposable objects in the theory, and recent work extends this notion from the classical case to the vector setting. A $PF(u)$4-parking function $PF(u)$5 is prime if

$PF(u)$6

If $PF(u)$7, every $PF(u)$8-parking function is prime. The corresponding sets are denoted $PF(u)$9 and $IPF(u)$0 for all and increasing prime $IPF(u)$1-parking functions (Armon et al., 2024).

This strictness has both combinatorial and geometric meaning. In the parking-process language, it replaces the threshold “at least $IPF(u)$2” by “strictly more than $IPF(u)$3.” In the lattice-path model, it means that the associated path touches the right boundary only at the endpoints, so the path is indecomposable with respect to the boundary (Armon et al., 2024).

A particularly useful reformulation is the modified-boundary criterion. If

$IPF(u)$4

then a sequence $IPF(u)$5 is a prime $IPF(u)$6-parking function if and only if it is a $IPF(u)$7-parking function. Equivalently,

$IPF(u)$8

A second characterization states that if $IPF(u)$9, then $u$0 if and only if removing any entry of $u$1 that is $u$2 produces a $u$3-parking function of length $u$4 (Armon et al., 2024).

Primeness is compatible with a unique direct-sum decomposition. Define

$u$5

If a $u$6-parking path $u$7 meets these special points in

$u$8

then $u$9 decomposes as

$x$00

where each $x$01 is prime with respect to an induced boundary vector $x$02. After sorting a $x$03-parking function and applying the same path decomposition, one obtains a unique direct-sum decomposition

$x$04

This is the vector-parking-function analogue of the classical prime decomposition (Armon et al., 2024).

For arithmetic progressions $x$05, primeness admits closed formulas. The number of prime $x$06-parking functions is

$x$07

and the increasing prime count is

$x$08

When $x$09, these reduce to

$x$10

and

$x$11

the standard classical enumerations (Armon et al., 2024).

5. Relation to graphical, matrix, and hypergraph parking functions

A persistent point of clarification in the literature is that vector parking functions are not merely another presentation of graphical parking functions. For a connected multigraph $x$12 with root $x$13, a $x$14-parking function is defined by cut conditions on all nonempty subsets. One paper proves that if $x$15 is invariant under the natural $x$16-action, then $x$17 must be one of three types: an $x$18-tree, an $x$19-cycle, or $x$20. In those cases,

$x$21

for the corresponding special vectors

$x$22

If $x$23 is not symmetric under permutations of coordinates, then it cannot equal $x$24 for any $x$25. This sharply limits the overlap between the graphical and vector theories (Gaydarov et al., 2015).

The matrix-theoretic generalization goes still further. For an integer matrix $x$26 with $x$27, off-diagonal entries $x$28, and an admissible cone $x$29 of positive vectors $x$30 with $x$31, one defines $x$32-parking functions by requiring that for every nonzero $x$33, there exists $x$34 with $x$35 such that

$x$36

A central theorem shows that the set $x$37 is independent of $x$38; one therefore speaks simply of $x$39-parking functions. These are in bijection with $x$40-recurrent configurations via

$x$41

and both sets have cardinality

$x$42

This framework simultaneously generalizes classical parking functions, $x$43-parking functions, and recurrent sandpile configurations (Ma et al., 2014).

Hypergraph parking functions give another extension. For a connected hypergraph $x$44 with sink $x$45, an $x$46-parking function is a vector $x$47 such that for every nonempty subset $x$48, there exists $x$49 with

$x$50

where $x$51 counts hyperedges containing $x$52 that are not entirely contained in $x$53. In complete hypergraphs $x$54, these objects coincide with explicit $x$55-parking functions, so the vector theory appears here as an exact specialization of a broader hypergraph chip-firing theory (Blanton et al., 13 Aug 2025).

6. Higher-dimensional, restricted, and probabilistic developments

The vector framework has several significant extensions. One is the 2-dimensional $x$56-parking-function theory, where one studies pairs

$x$57

whose order statistics are bounded along a lattice path in a monotone node set

$x$58

For affine node sets of the form

$x$59

the theory distinguishes three cases: a diagonal or independent case realized by a disjoint union of weighted complete graphs, a symmetric mixed case $x$60 realized by a complete graph with a 5-parameter weight pattern, and an asymmetric mixed case $x$61, for which there is no graph $x$62 satisfying

$x$63

More generally, if $x$64 is $x$65-invariant, then there exists a 2-dimensional node set $x$66 with

$x$67

This is a precise higher-dimensional analogue of the overlap between vector and graphical parking functions (Snider et al., 2023).

Another direction is preference restriction. For a subset $x$68, one defines

$x$69

If

$x$70

then $x$71-restricted parking functions are in bijection with $x$72-parking functions. This unifies several variants: defective parking with fewer spots than cars, prime parking functions, and parking with multiple-car capacities. In particular, prime parking functions on $x$73 cars correspond to

$x$74

and more generally prime restricted parking functions reduce to ordinary restricted parking functions by a shift of allowed preferences (Bown et al., 15 Jul 2025).

Refined statistics have also been extended to the vector setting. In the lucky-car theory, a car is lucky if it parks in its preferred spot. For a weakly increasing vector

$x$75

the outcome of a $x$76-parking function is an ordered set partition

$x$77

with $x$78's marking unavailable spots. One paper characterizes precisely which outcomes realize a fixed lucky set and gives formulas both for outcomes and for the number of $x$79-parking functions themselves with a fixed lucky set or fixed number of lucky cars (Ferreri et al., 19 Aug 2025).

There are also weighted and asymptotic refinements. For

$x$80

of length $x$81, the $x$82-enumerator by number of $x$83's is

$x$84

Under the congruence condition $x$85, the same paper gives a $x$86-analogue for $x$87-parking functions as a sum over admissible block sizes in $x$88 (Yang, 2024).

Finally, for random $x$89-parking functions with

$x$90

the asymptotic regime depends sharply on whether $x$91 or $x$92. In the generic case $x$93, mixed moments of coordinates have uniform-like leading terms and the covariance of two coordinates is asymptotically negative. In the special case $x$94, moment asymptotics, boundary laws, and displacement fluctuations change scale, and the paper explicitly emphasizes that the asymptotic scenario in the generic situation $x$95 is in sharp contrast with that of the special situation $x$96 (Yin, 2021).

Taken together, these developments show that vector parking functions form a coherent but nonuniformly notated domain linking order-statistics inequalities, parking processes with capacities, Catalan-style path models, tree and forest bijections, prime factorization phenomena, and several broader frameworks—graphical, matrix, hypergraph, restricted, and higher-dimensional—within which the vector theory is sometimes an exact specialization and sometimes only a limited overlap.