Parking functions, valet functions and priority queues

Gilbey, Julian; Kalikow, Louis H.

doi:10.1016/s0012-365x(99)90085-7

Cited by 18 publications

(12 citation statements)

References 14 publications

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“…Section 5 gives a characterization for compatible pairs in terms of pattern-avoidance. We demonstrate that compatible pairs are essentially allowable pairs introduced in [4] and investigated further in [3,18,22]. To answer our original question, we show in Corollary 5.8 that every allowable pair can be obtained by standardizing the entries in the last two columns of some standard composition tableau.…”

Section: Introductionmentioning

confidence: 88%

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

Tewari

Willigenburg

2019

Journal of Combinatorial Theory, Series A

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We introduce a generalization of semistandard composition tableaux called permuted composition tableaux. These tableaux are intimately related to permuted basement semistandard augmented fillings studied by Haglund, Mason and Remmel. Our primary motivation for studying permuted composition tableaux is to enumerate all possible ordered pairs of permutations (σ 1 , σ 2 ) that can be obtained by standardizing the entries in two adjacent columns of an arbitrary composition tableau. We refer to such pairs as compatible pairs. To study compatible pairs in depth, we define a 0-Hecke action on permuted composition tableaux. This action naturally defines an equivalence relation on these tableaux. Certain distinguished representatives of the resulting equivalence classes in the special case of two-columned tableaux are in bijection with compatible pairs. We provide a bijection between two-columned tableaux and labeled binary trees. This bijection maps a quadruple of descent statistics for 2-columned tableaux to left and right ascent-descent statistics on labeled binary trees introduced by Gessel, and we use it to prove that the number of compatible pairs is (n + 1) n−1 . 14 4.2. Bijection between LD n and SPCT((2 n )) 15 4.3. Plane binary trees, unlabeled and labeled 162010 Mathematics Subject Classification. Primary 05E10, 20C08; Secondary 05A05, 05A19, 05C20, 05E05, 05E15.

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Section: Introductionmentioning

confidence: 88%

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

Tewari

Willigenburg

2019

Journal of Combinatorial Theory, Series A

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“…For BFS version I, as mentioned in the proof, the queue length at time k coincides with the number of cars y k (π) that attempt to park at spot k (whether successful or not), and the number of new vertices in the queue at time k coincides with the number of cars r k (π) whose first preference is spot k. We have For version II of the BFS algorithm, the queue is empty at times 5, 6, 8, separating the 12 available spots into disjoint segments of [1,4], [7], [9,12], with respective length 4, 1, 4. This coincides with the number of non-root vertices in the left-to-right trees of the rooted forest.…”

Section: 2mentioning

confidence: 99%

Parking functions: From combinatorics to probability

Kenyon¹,

Yin²

2021

Preprint

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Suppose that m drivers each choose a preferred parking space in a linear car park with n spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If all drivers park successfully, the sequence of choices is called a parking function. Classical parking functions correspond to the case m = n; we study here combinatorial and probabilistic aspects of this generalized case.We construct a family of bijections between parking functions PF(m, n) with m cars and n spots and spanning forests F (n + 1, n + 1 − m) with n + 1 vertices and n + 1 − m distinct trees having specified roots. This leads to a bijective correspondence between PF(m, n) and monomial terms in the associated Tutte polynomial of a disjoint union of n − m + 1 complete graphs. We present an identity between the "inversion enumerator" of spanning forests with fixed roots and the "displacement enumerator" of parking functions. The displacement is then related to the number of graphs on n + 1 labeled vertices with a fixed number of edges, where the graph has n + 1 − m disjoint rooted components with specified roots.We investigate various probabilistic properties of a uniform parking function, giving a formula for the law of a single coordinate. As a side result we obtain a recurrence relation for the displacement enumerator. Adapting known results on random linear probes, we further deduce the covariance between two coordinates when m = n.

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“…Parking functions have been found in connection to many other combinatorial structures such as acyclic mappings, polytopes, non-crossing partitions, non-nesting partitions, hyperplane arrangements, etc. Refer to [6,5,7,11,14,15] for more information.…”

Section: Introductionmentioning

confidence: 99%

Tutte polynomials and G-parking functions

Chang

Yeh

2010

Advances in Applied Mathematics

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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.

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Parking functions, valet functions and priority queues

Cited by 18 publications

References 14 publications

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

Parking functions: From combinatorics to probability

Tutte polynomials and G-parking functions

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