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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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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).
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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Cited by 15 (0 self)
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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 ...
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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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...
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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Cited by 14 (0 self)
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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.
Conjectured combinatorial models for the Hilbert series of generalized diagonal harmonics modules
 Electron.J.Combin.11 (2004), R68; 64
"... Haglund and Loehr previously conjectured two equivalent combinatorial formulas for the Hilbert series of the GarsiaHaiman diagonal harmonics modules. These formulas involve weighted sums of labelled Dyck paths (or parking functions) relative to suitable statistics. This article introduces a third c ..."
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Cited by 6 (4 self)
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Haglund and Loehr previously conjectured two equivalent combinatorial formulas for the Hilbert series of the GarsiaHaiman diagonal harmonics modules. These formulas involve weighted sums of labelled Dyck paths (or parking functions) relative to suitable statistics. This article introduces a third combinatorial formula that is shown to be equivalent to the first two. We show that the four statistics on labelled Dyck paths appearing in these formulas all have the same univariate distribution, which settles an earlier question of Haglund and Loehr. We then introduce analogous statistics on other collections of labelled lattice paths contained in trapezoids. We obtain a fermionic formula for the generating function for these statistics. We give bijective proofs of the equivalence of several forms of this generating function. These bijections imply that all the new statistics have the same univariate distribution. Using these new statistics, we conjecture combinatorial formulas for the Hilbert series of certain generalizations of the diagonal harmonics modules. 1
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 ...
kflaw Preference Sets
, 806
"... In this paper, let Pl n;≤s;k denote a set of kflaw preference sets (a1,...,an) with n parking spaces satisfying that 1 ≤ ai ≤ s for any i and a1 = l and pl n;≤s;k = Pl n;≤s;k . We use a combinatorial approach to the enumeration of kflaw preference sets by their leading terms. The approach relies ..."
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In this paper, let Pl n;≤s;k denote a set of kflaw preference sets (a1,...,an) with n parking spaces satisfying that 1 ≤ ai ≤ s for any i and a1 = l and pl n;≤s;k = Pl n;≤s;k . We use a combinatorial approach to the enumeration of kflaw preference sets by their leading terms. The approach relies on bijections between the kflaw preference sets and labeled rooted forests. Some bijective results between certain sets of kflaw preference sets of distinct leading terms are also given. We derive some formulas and recurrence relations for the sequences p l n;≤s;k and give the generating functions for these sequences.