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Coalescent Random Forests
 J. COMBINATORIAL THEORY A
, 1998
"... Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ..."
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Cited by 38 (18 self)
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Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...
AbelCayleyHurwitz multinomial expansions associated with random mappings, forests, and subsets
, 1998
"... Extensions of binomial and multinomial formulae due to Abel, Cayley and Hurwitz are related to the probability distributions of various random subsets, trees, forests, and mappings. For instance, an extension of Hurwitz's binomial formula is associated with the probability distribution of the random ..."
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Cited by 13 (12 self)
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Extensions of binomial and multinomial formulae due to Abel, Cayley and Hurwitz are related to the probability distributions of various random subsets, trees, forests, and mappings. For instance, an extension of Hurwitz's binomial formula is associated with the probability distribution of the random set of vertices of a fringe subtree in a random forest whose distribution is defined by terms of a multinomial expansion over rooted labeled forests which generalizes Cayley's expansion over unrooted labeled trees. Contents 1 Introduction 2 Research supported in part by N.S.F. Grant DMS9703961 2 Probabilistic Interpretations 5 3 Cayley's multinomial expansion 11 4 Random Mappings 14 4.1 Mappings from S to S : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 4.2 The random set of cyclic points : : : : : : : : : : : : : : : : : : : : : : : 18 5 Random Forests 19 5.1 Distribution of the roots of a pforest : : : : : : : : : : : : : : : : : : : : 19 5.2 Conditioning on the set...
A bijective proof for the number of labelled qtrees
 Ars Combinatoria
, 1988
"... We giv e abijective proof that the number of vertex labeled qtrees on n vertices is given by ⎛ n ⎞ ⎝ q ⎠ [ qn − q2 n − q − 2 + 1] The bijection transforms each pair ( S, f) where S is a qelement subset of an nset, and f is a function mapping an ( n − q − 2)set to a ( qn − q 2 + 1)set into a la ..."
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Cited by 4 (0 self)
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We giv e abijective proof that the number of vertex labeled qtrees on n vertices is given by ⎛ n ⎞ ⎝ q ⎠ [ qn − q2 n − q − 2 + 1] The bijection transforms each pair ( S, f) where S is a qelement subset of an nset, and f is a function mapping an ( n − q − 2)set to a ( qn − q 2 + 1)set into a labeled qtree on n nodes by a cutandpaste process. As a special case, q = 1 yields a new bijective proof of Cayley’s formula for labeled trees. The general bijection also provides for the enumeration of labeled qtrees in which a specified subset of the vertices forms a clique. 1.
Enumeration of Some Labelled Trees
"... . In this paper we are interesting in the enumeration of rooted labelled trees according to the relationship between the root and its sons. Let Tn;k be the family of Cayley trees on [n] such that the root has exactly k smaller sons. In a first time we give a bijective proof of the fact that jTn+1;k ..."
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Cited by 3 (0 self)
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. In this paper we are interesting in the enumeration of rooted labelled trees according to the relationship between the root and its sons. Let Tn;k be the family of Cayley trees on [n] such that the root has exactly k smaller sons. In a first time we give a bijective proof of the fact that jTn+1;k j = \Gamma n k \Delta n n\Gammak . Moreover, we use the family Tn+1;0 of Cayley trees for which the root is smaller than all its sons to give combinatorial explanations of various identities involving n n . We rely this family to the enumeration of minimal factorization of the ncycle (1; 2; : : : ; n) as a product of transpositions. Finally, we use the fact that jTn+1;0 j = n n to prove bijectively that there are 2n n ordered alternating trees on [n + 1]. R' esum' e. Dans cet article nous nous int'eressons `a l"enum'eration d'arbres 'etiquet'es enracin'es, en consid'erant un nouveau param`etre relatif `a l'ordre existant entre la racine et ses fils. Soit donc Tn;k la famille d...
Random Trees
 J. Logic Comp
, 2000
"... We briefly review a small fraction of the literature for anyone interested in random trees in logic, linguistics, or computer science. This note should be regarded as a sort of disorganized bibliography for anyone who is interested in pursuing the subject. One way to study languages, algorithms, ..."
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Cited by 1 (1 self)
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We briefly review a small fraction of the literature for anyone interested in random trees in logic, linguistics, or computer science. This note should be regarded as a sort of disorganized bibliography for anyone who is interested in pursuing the subject. One way to study languages, algorithms, and similar mechanical constructions is to choose some inputs at random, feed them into the machine, and see what happens. We can do this theoretically as follows: suppose that each input is either ACCEPTED or REJECTED. Since we are often interested in quite large inputs, we might ask: Question 1 For each n, let I n be the set of all inputs of size n. 1. Given a machine M , what is the probability p n that a randomly chosen input from I n is accepted? 2. What happens to p n as n gets large? Question 1(1), "what is p n ?" is often a quite difficult problem in "enumerative combinatorics. " On the other hand, Question 1(2), "what happens to p n as n !1?" is often tractible. The answer to t...
On a Basis for the Framed Link Vector Space Spanned by Chord Diagrams
, 2008
"... Abstract In view of the result of Kontsevich, [5] now often called “the fundamental theorem of Vassiliev theory”, identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the “4Trelation ” ..."
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Abstract In view of the result of Kontsevich, [5] now often called “the fundamental theorem of Vassiliev theory”, identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the “4Trelation ” (and in the unframed case, originally considered in [7], [5], and [1], the “1T ” or “isolated chord relation”), it is a problem of some interest to provide a basis for the space of chord diagrams modulo the 4Trelation. We construct the basis for the vector space spanned by chord diagrams with n chords and m link components, modulo 4T relations for n ≤ 5. 1
The enumeration of labelled spanning trees of Km,n
 AUSTRALASIAN JOURNAL OF COMBINATORICS VOLUME 28 (2003), PAGES 73–79
, 2003
"... Using a bijection to decompose a labelled rooted bipartite tree into several ones with smaller size and their exponential generating functions, this paper concerns the number of labelled spanning trees of the complete bipartite graph Km,n. ..."
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Using a bijection to decompose a labelled rooted bipartite tree into several ones with smaller size and their exponential generating functions, this paper concerns the number of labelled spanning trees of the complete bipartite graph Km,n.