The cluster complex on one hand, parking functions on the other hand, are two combinatorial (po)sets that can be associated to a finite real reflection group. Cluster parking functions are obtained by taking an appropriate fiber product (over noncrossing partitions). There is a natural structure of simplicial complex on these objects, and our main goal is to show that it has the homotopy type of a (pure) wedge of spheres. The unique nonzero homology group (as a representation of the underlying reflection group) is a sign-twisted parking representation, which is the same as Gordon's quotient of diagonal coinvariants. Along the way, we prove some properties of the poset of parking functions. We also provide a long list of remaining open problems.
We construct an action of the braid group on $n$ strands on the set of parking functions of $n$ cars such that elementary braids have orbits of length 2 or 3. The construction is motivated by a theorem of Lyashko and Looijenga stating that the number of the distinguished bases for $A_n$ singularity equals $(n+1)^{n-1}$ and thus equals the number of parking functions. We construct an explicit bijection between the set of parking functions and the set of distinguished bases, which allows us to translate the braid group action on distinguished bases in terms of parking functions.
We test the umbral methods introduced by Rota and Taylor within the theory of representation of symmetric group. We define a simple bijection between the set of all parking functions of length $n$ and the set of all noncrossing partitions of $\{1,2,...,n\}$. Then we give an umbral expression of the Frobenius characteristic of the parking function module introduced by Haiman that allows an explicit relation between this symmetric function and the volume polynomial of Pitman and Stanley.
We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund's bijection exchanging the pairs of statistics (area,dinv) and (bounce,area) on Dyck paths, and Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constru…
A pair $(G,K)$ of a group and its subgroup is called a Gelfand pair if the induced trivial representation of $K$ on $G$ is multiplicity free. Let $(a_j)$ be a sequence of positive integers of length $n$, and let $(b_i)$ be its non-decreasing rearrangement. The sequence $(a_i)$ is called a parking function of length $n$ if $b_i \leq i$ for all $i=1,\...,n$. In this paper we study certain Gelfand pairs in relation with parking functions. In particular, we find explicit descriptions of the decomposition of the associated induced trivial representations into irreducibles. We obtain and study a new…
Classical parking functions can be defined in terms of drivers with preferred parking spaces searching a linear parking lot for an open parking spot. We may consider this linear parking lot as a collection of $n$ vertices (parking spots) arranged in a directed path. We generalize this notion to allow for more complicated "parking lots" and define parking functions on arbitrary directed graphs. We then consider a relationship proved by Lackner and Panholzer between parking functions on trees and "mapping digraphs" and we show that a similar relationship holds when edge orientations are reversed…
Parking functions were classically defined for $n$ cars attempting to park on a one-way street with $n$ parking spots, where cars only drive forward. Subsequently, parking functions have been generalized in various ways, including allowing cars the option of driving backward. The set $PF_{n,k}$ of $k$-Naples parking functions have cars who can drive backward a maximum of $k$ steps before driving forward. A recursive formula for $|PF_{n,k}|$ has been obtained, though deriving a closed formula for $|PF_{n,k}|$ appears difficult. In addition, an important subset $B_{n,k}$ of $PF_{n,k}$, called th…
We introduce several associative algebras and series of vector spaces associated to these algebras. Using lattice vertex operators, we obtain dimension and character formulae for these spaces. In particular, we a series of representations of symmetric groups which turn out to be isomorphic to parking function modules. We also construct series of vector spaces whose dimensions are Catalan numbers and Fuss--Catalan numbers respectively. Conjecturally, these spaces are related to spaces of global sections of vector bundles on (zero fibres of) Hilbert schemes and representations of rational Chered…
A parking function of length $n$ is prime if we obtain a parking function of length $n-1$ by deleting one 1 from it. In this note we give a new direct proof that the number of prime parking functions of length $n$ is $(n-1)^{n-1}$. This proof leads to a new interpretation, in close terms to the definition of parking function.
We prove that the combinatorial side of the "Rational Shuffle Conjecture" provides a Schur-positive symmetric polynomial. Furthermore, we prove that the contribution of a given rational Dyck path can be computed as a certain skew LLT polynomial, thus generalizing the result of Haglund, Haiman, Loehr, Remmel and Ulyanov. The corresponding skew diagram is described explicitly in terms of a certain (m,n)-core.
Classical parking functions are defined as the parking preferences for $n$ cars driving (from west to east) down a one-way street containing parking spaces labeled from $1$ to $n$ (from west to east). Cars drive down the street toward their preferred spot and park there if the spot is available. Otherwise, the car continues driving down the street and takes the first available parking space, if such a space exists. If all cars can park using this parking rule, we call the $n$-tuple containing the cars' parking preferences a parking function. In this paper, we introduce a generalization of the …
In 1980, G. Kreweras gave a recursive bijection between forests and parking functions. In this paper we construct a nonrecursive bijection from forests onto parking functions, which answers a question raised by R. Stanley. As a by-product, we obtain a bijective proof of Gessel and Seo's formula for lucky statistic on parking functions.
We study the enumeration problem for different kind of tree parking functions introduced recently, called tree parking functions, tree parking distributions, prime tree parking functions, and prime tree parking distributions, for rooted labelled trees of important combinatorial tree families including labelled ordered, unordered and binary trees. Using combinatorial decompositions of the underlying structures yields, after solving the resulting equations, implicit characterizations of suitable generating functions of the total number of such tree parking functions for trees of size $n$ and $n$…
We introduce a new parking procedure called MVP parking in which $n$ cars sequentially enter a one-way street with a preferred parking spot from the $n$ parking spots on the street. If their preferred spot is empty, they park there. Otherwise, they park there and the car parked in that spot is bumped to the next unoccupied spot on the street. If all cars can park under this parking procedure, we say the list of preferences of the $n$ cars is an MVP parking function of length $n$. We show that the set of (classical) parking functions is exactly the set of MVP parking functions although the park…
Let $G$ be a connected graph with vertex set $\{0,1,2,...,n\}$. We allow $G$ to have multiple edges and loops. In this paper, we give a characterization of external activity by some parameters of $G$-parking functions. In particular, we give the definition of the bridge vertex of a $G$-parking function and obtain an expression of the Tutte polynomial $T_G(x,y)$ of $G$ in terms of $G$-parking functions. We find the Tutte polynomial enumerates the $G$-parking function by the number of the bridge vertices.
We give a combinatorial formula for LLT cumulants of unicellular unilevelled shapes in terms of parking functions and Cayley trees. Our formula implies previously conjectured Schur positivity.
This paper studies a generalization of parking functions named $k$-Naples parking functions, where backward movement is allowed. One consequence of backward movement is that the number of ascending $k$-Naples is not the same as the number of descending $k$-Naples. This paper focuses on generalizing the bijections of ascending parking functions with combinatorial objects enumerated by the Catalan numbers in the setting of both ascending and descending $k$-Naples parking functions. These combinatorial objects include Dyck paths, binary trees, triangulations of polygons, and non-crossing partitio…
We give an exact enumerative formula for the minimal acyclic deterministic finite automata. This formula is obtained from a bijection between a family of generalized parking functions and the transitions functions of acyclic automata.
Parking functions, classically defined in terms of cars with preferred parking spots on a directed path attempting to park there, arise in many combinatorial situations and have seen various generalizations. In particular, parking functions have been defined for general digraphs, which yields many more enumeration problems. For example, in a directed tree whose edges are orientated away from the root, it is unknown in general how the number of parking functions on it changes once the orientation is reversed, even in the case when the tree is a star. We show that this orientation reversal resul…
We apply the concept of parking functions to rooted labelled trees and functional digraphs of mappings (i.e., functions $f : [n] \to [n]$) by considering the nodes as parking spaces and the directed edges as one-way streets: Each driver has a preferred parking space and starting with this node he follows the edges in the graph until he either finds a free parking space or all reachable parking spaces are occupied. If all drivers are successful we speak about a parking function for the tree or mapping. We transfer well-known characterizations of parking functions to trees and mappings. Especial…
In this paper, we view parking functions viewed as labeled Dyck paths in order to study a notion of pattern avoidance first introduced by Remmel and Qiu. In particular we enumerate the parking functions avoiding any set of two or more patterns of length 3, and we obtain a number of well-known combinatorial sequences as a result. Along the way, we find bijections between specific sets of pattern-avoiding parking functions and a number of combinatorial objects such as partitions of polygons and trees with certain restrictions.
For a labeled, rooted tree with edges oriented towards the root, we consider the vertices as parking spots and the edge orientation as a one-way street. Each driver, starting with her preferred parking spot, searches for and parks in the first unoccupied spot along the directed path to the root. If all $n$ drivers park, the sequence of spot preferences is called a parking function. We consider the sequences, called \emph{prime} parking functions, for which each driver parks and each edge in the tree is traversed by some driver after failing to park at her preferred spot. We prove that the tota…
A depth-first search version of Dhar's burning algorithm is used to give a bijection between the parking functions of a graph and labeled spanning trees, relating the degree of the parking function with the number of inversions of the spanning tree. Specializing to the complete graph answers a problem posed by R. Stanley.
We enumerate interlaced pairs of parking functions whose underlying Dyck path has a bounded height. We obtain an explicit formula for this enumeration in the form of a quotient of analogs of Chebicheff polynomials having coefficients in the ring of symmetric functions.
Kreweras proved that the reversed sum enumerator for parking functions of length $n$ is equal to the inversion enumerator for labeled trees on $n+1$ vertices. Recently, Perkinson, Yang, and Yu gave a bijective proof of this equality that moreover generalizes to graphical parking functions. Using a depth-first search variant of Dhar's burning algorithm they proved that the codegree enumerator for $G$-parking functions equals the $κ$-number enumerator for spanning trees of $G$. The $κ$-number is a kind of generalized tree inversion number originally defined by Gessel. We extend the work of Perki…
It is known that the Pak-Stanley labeling of the Shi hyperplane arrangement provides a bijection between the regions of the arrangement and parking functions. For any graph G, we define the G-semiorder arrangement and show that the Pak-Stanley labeling of its regions produces all G-parking functions.
We give a recursive definition of generalized parking function that allows us to view them as a species. From there we compute a non-commutative characteristic of the generalized parking function module, and deduce some enumeration formulas of structures and isomorphism types. We give as well an interpretation in several bases of non-commutative symmetric functions. Finally, we investigate an inclusion-exclusion formula given by Kung and Yan.
The Dyck path algebra construction of Carlsson and Mellit from arXiv:1508.06239 is interpreted as a representation of "the positive part" of the group of toric braids. Then certain sums over $(m,n)$-parking functions are related to evaluations of this representation on some special braids. The compositional $(km,kn)$-shuffle conjecture of Bergeron, Garsia, Leven and Xin from arXiv:1404.4616 is then shown to be a corollary of this relation.
Parking sequences (a generalization of parking functions) are defined by specifying car lengths and requiring that a car attempts to park in the first available spot after its preference. If it does not fit there, then a collision occurs and the car fails to park. In contrast, parking assortments generalize parking sequences (and parking functions) by allowing cars (also of assorted lengths) to seek forward from their preference to identify a set of contiguous unoccupied spots in which they fit. We consider both parking sequences and parking assortments and establish that the number of prefere…
We study Schroder paths drawn in a (m,n) rectangle, for any positive integers m and n. We get explicit enumeration formulas, closely linked to those for the corresponding (m,n)-Dyck paths. Moreover we study a Schroder version of (m,n)-parking functions, and associated (q,t)-analogs.
The notion of parking sequences is a new generalization of parking functions introduced by Ehrenborg and Happ. In the parking process defining the classical parking functions, instead of each car only taking one parking space, we allow the cars to have different sizes and each takes up a number of adjacent parking spaces after a trailer $T$ parked on the first $z-1$ spots. A preference sequence in which all the cars are able to park is called a parking sequence. In this paper, we study increasing parking sequences and count them via bijections to lattice paths with right boundaries. Then we st…
We introduce type C parking functions, encoded as vertically labelled lattice paths and endowed with a statistic dinv'. We define a bijection from type C parking functions to regions of the Shi arrangement of type C, encoded as diagonally labelled ballot paths and endowed with a natural statistic area'. This bijection is a natural analogue of the zeta map of Haglund and Loehr and maps dinv' to area'. We give three different descriptions of it.
The shuffle conjecture expresses a relationship between parking functions, diagonal harmonics, and the Bergeron-Garsia $\nabla$ operator. Recent conjectures about a family of modified Hall-Littlewood operators made by Haglund, Morse, and Zabrocki sharpen the shuffle conjecture and suggest a variety of combinatorial properties of parking functions. In particular, their conjectures combined with previously established commutativity laws of the Hall-Littlewood operators, suggest the existence of certain bijections relating parking functions with different diagonal compositions. In this paper we f…
