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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...
Random mappings, forests, and subsets associated with AbelCayleyHurwitz multinomial expansions
, 2001
"... Various random combinatorial objects, such as mappings, trees, forests, and subsets of a finite set, are constructed with probability distributions related to the binomial and multinomial expansions due to Abel, Cayley and Hurwitz. Relations between these combinatorial objects, such as Joyal's b ..."
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Cited by 13 (9 self)
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Various random combinatorial objects, such as mappings, trees, forests, and subsets of a finite set, are constructed with probability distributions related to the binomial and multinomial expansions due to Abel, Cayley and Hurwitz. Relations between these combinatorial objects, such as Joyal's bijection between mappings and marked rooted trees, have interesting probabilistic interpretations, and applications to the asymptotic structure of large random trees and mappings. 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. Research supported in part by N.S.F. Grants DMS 9703961 and DMS0071448 1 Contents 1
The Multinomial Distribution on Rooted Labeled Forests
, 1997
"... For a probability distribution (p s ; s 2 S) on a finite set S, call a random forest F of rooted trees labeled by S (with edges directed away from the roots) a pforest if given F has m edges the vector of outdegrees of vertices of F has a multinomial distribution with parameters m and (p s ; s 2 ..."
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Cited by 10 (10 self)
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For a probability distribution (p s ; s 2 S) on a finite set S, call a random forest F of rooted trees labeled by S (with edges directed away from the roots) a pforest if given F has m edges the vector of outdegrees of vertices of F has a multinomial distribution with parameters m and (p s ; s 2 S), and given also these outdegrees the distribution of F is uniform on all forests with the given outdegrees. The family of distributions of pforests is studied, and shown to be closed under various operations involving deletion of edges. Some related enumerations of rooted labeled forests are obtained as corollaries. 1 Introduction Let F(S) denote the set of all forests of rooted trees labeled by a finite set S of size jSj. Each f 2 F(S) is a directed graph labeled by S, that is a subset of S \Theta S, such that each Research supported in part by N.S.F. Grant DMS9703961 connected component of the graph is a tree with edges directed away from some root vertex. The notation v f ...
SIMPLY GENERATED TREES, CONDITIONED GALTON–WATSON TREES, RANDOM ALLOCATIONS AND CONDENSATION (EXTENDED ABSTRACT)
, 2012
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A Poisson * negative binomial convolution law for random polynomials over finite fields
, 1998
"... Let F q [X ] denote a polynomial ring over a finite field F q with q elements. Let Pn be the set of monic polynomials over F q of degree n. Assuming that each of the q possible monic polynomials in Pn is equally likely, we give a complete characterization of the limiting behavior ofP(\Omega n = m ..."
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Cited by 2 (2 self)
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Let F q [X ] denote a polynomial ring over a finite field F q with q elements. Let Pn be the set of monic polynomials over F q of degree n. Assuming that each of the q possible monic polynomials in Pn is equally likely, we give a complete characterization of the limiting behavior ofP(\Omega n = m) as n !1 by a uniform asymptotic formula valid for m 1 and n \Gamma m !1, n represents the number (multiplicities counted) of irreducible factors in the factorization of a random polynomial in Pn . The distribution n is essentially the convolution of a Poisson distribution with mean log n and a negative binomial distribution with parameters q and q . Such a convolution law exhibits three modes of asymptotic behaviors: when m is small, it behaves like a Poisson distribution; when m becomes large, its behavior is dominated by a negative binomial distribution, the transitional behavior being essentially a parabolic cylinder function (or some linear combinations of the standard normal law and its iterated integrals). As applications of this uniform asymptotic formula, we derive most known results concerningP(\Omega n = m) and present many new ones like the unimodality of the distribution. The methods used are widely applicable to other problems on multiset constructions. An extension to R'enyi's problem, concerning the distribution of the difference of the (total) number of irreducibles and the number of distinct irreducibles, is also presented. AMS 1991 Mathematics subject classification: Primary 11T06; secondary 60C05.