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Phase transition for parking blocks, Brownian excursion and coalescence
, 2005
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A family of bijections between Gparking functions and spanning trees
 J. Combin. Theory Ser. A
, 2005
"... Abstract. For a directed graph G on vertices {0, 1,..., n}, a Gparking function is an ntuple (b1,..., bn) of nonnegative integers such that, for every nonempty subset U ⊆ {1,..., n}, there exists a vertex j ∈ U for which there are more than bj edges going from j to G − U. We construct a family o ..."
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Abstract. For a directed graph G on vertices {0, 1,..., n}, a Gparking function is an ntuple (b1,..., bn) of nonnegative integers such that, for every nonempty subset U ⊆ {1,..., n}, there exists a vertex j ∈ U for which there are more than bj edges going from j to G − U. We construct a family of bijective maps between the set PG of Gparking functions and the set TG of spanning trees of G rooted at 0, thus providing a combinatorial proof of PG  = TG. 1.
Parking Functions, Empirical Processes, and the Width of Rooted Labeled Trees
"... This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking f ..."
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Cited by 21 (6 self)
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This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking functions and the empirical processes of mathematical statistics. Our result turns out to be a consequence of the strong convergence of empirical processes to the Brownian bridge (Komlos, Major and Tusnady, 1975).
Generalized Parking Functions, Tree Inversions and Multicolored Graphs
"... A generalized xparking function associated to x = (a, b, b, . . . , b) # N n is a sequence (a 1 , a 2 , . . . , an ) of positive integers whose nondecreasing rearrangement b 1 # b 2 # # b n satisfies b i # a + (i  1)b. The set of xparking functions is equinumerate with the set of se ..."
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Cited by 17 (1 self)
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A generalized xparking function associated to x = (a, b, b, . . . , b) # N n is a sequence (a 1 , a 2 , . . . , an ) of positive integers whose nondecreasing rearrangement b 1 # b 2 # # b n satisfies b i # a + (i  1)b. The set of xparking functions is equinumerate with the set of sequences of rooted bforests on [n]. We construct a bijection between these two sets. We show that the sum enumerator of complements of xparking functions is identical to the inversion enumerator of sequences of rooted bforests by generating function analysis. Combinatorial correspondences between the sequences of rooted forests and xparking functions is also given in terms of depthfirst search and breadthfirst search on multicolored graphs. 1 Introduction The notion of parking function was introduced by Konheim and Weiss as a colorful way to study a hashing problem. In the paper [9], they proved the number of parking functions of length n is (n + 1) n1 . Later the subject has attra...
On the Enumeration of Generalized Parking Functions
, 1999
"... Let x = (x1 , x2 , . . . , xn) # N n . Define a xparking function to be a sequence (a1 , a2 , . . . , an) of positive integers whose nondecreasing rearrangement b1 # b2 # # bn satisfies b i # x1 + +x i . Let Pn (x) denote the number of xparking functions. We discuss the enumeration ..."
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Let x = (x1 , x2 , . . . , xn) # N n . Define a xparking function to be a sequence (a1 , a2 , . . . , an) of positive integers whose nondecreasing rearrangement b1 # b2 # # bn satisfies b i # x1 + +x i . Let Pn (x) denote the number of xparking functions. We discuss the enumerations of such generalized parking functions. In particular, We give the explicit formulas and present two proofs, one by combinatorial argument, and one by recurrence, for the number of xparking functions for x = (a, nm1 z } { b, . . . , b, c, m1 z } { 0, . . . , 0) and x = (a, nm1 z } { b, . . . , b, m z } { c, . . . , c). 1 Introduction The notion of parking function was introduced by Konheim and Weiss [3] in their studying of an occupancy problem in computer science. Then the subject has been studied by numerous mathematicians. For example, Foata and Riordan [1], and Francon [2] constructed bijections from the set of parking functions to the set of acyclic functions on ...
Enumeration Of Trees And One Amazing Representation Of The Symmetric Group
 University of Minnesota
, 1995
"... . In this paper we present a systematic approach to enumeration of different classes of trees and their generalizations. The principal idea is finding a bijection between these trees and some classes of Young diagrams or Young tableaux. The latter arise from the remarkable representation of the symm ..."
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. In this paper we present a systematic approach to enumeration of different classes of trees and their generalizations. The principal idea is finding a bijection between these trees and some classes of Young diagrams or Young tableaux. The latter arise from the remarkable representation of the symmetric group studied by Haiman in connection with diagonal harmonics (see [7]). Define the vector space V ' hx a 1 oe(1) : : : x an oe(n) joe 2 Sn ; 0 a i i \Gamma 1; 1 i ni. Let the symmetric group Sn act on Vn by the permutation of variables. It is known that dim(Vn ) = (n + 1) n\Gamma1 is equal to the number of labeled trees, and dim(Vn ) Sn = 1 n+1 \Gamma 2n n \Delta is equal to the number of plane trees with n vertices. There are combinatorial interpretations for the other multiplicities. We generalize all the results in case of kdimensional trees and (k + 1)ary trees. 1. Introduction. Define a vector space (11) V ' hx a1 oe(1) : : : x an oe(n) j oe 2 Sn ; 0 ...
HACHAGE, ARBRES, CHEMINS & GRAPHES
"... Mathématiques discrètes et continues se rencontrent et se complètent volontiers harmonieusement. C’est cette thèse que nous voudrions illustrer en discutant un problème classique aux ramifications nombreuses—l’analyse du hachage avec essais linéaires. L’exemple est issu de l’analyse d’algorithmes, d ..."
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Mathématiques discrètes et continues se rencontrent et se complètent volontiers harmonieusement. C’est cette thèse que nous voudrions illustrer en discutant un problème classique aux ramifications nombreuses—l’analyse du hachage avec essais linéaires. L’exemple est issu de l’analyse d’algorithmes, domaine fondé par Knuth et qui se situe luimême “à cheval ” entre l’informatique, l’analyse combinatoire, et la théorie des probabilités. Lors de son traitement se croisent au fil du temps des approches très diverses, et l’on rencontrera des questions posées par Ramanujan à Hardy en 1913, un travail d’été de Knuth datant de 1962 et qui est à l’origine de l’analyse d’algorithmes en informatique, des recherches en analyse combinatoire du statisticien Kreweras, diverses rencontres avec les modèles de graphes aléatoires au sens d’Erdös et Rényi, un peu d’analyse complexe et d’analyse asymptotique, des arbres qu’on peut voir comme issus de processus de GaltonWatson particuliers, et, pour finir, un peu de processus, dont l’ineffable mouvement Brownien! Tout ceci contribuant in fine à une compréhension très précise d’un modèle simple d’aléa discret.