Abel-Cayley-Hurwitz multinomial expansions associated with random mappings, forests, and subsets

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 de ned by terms of a multinomial expansion over rooted labeled forests which generalizes Cayley's expansion over unrooted labeled trees.

[1]  Jean Françon.  Preuves combinatoires des identites d'Abel , 1974, Discret. Math..

[2]  Jim Pitman,et al.  Enumerations of trees and forests related to branching processes and random walks , 1997, Microsurveys in Discrete Probability.

[3]  Louis W. Shapiro.  Voting blocks, reluctant functions, and a formula of Hurwitz , 1991, Discret. Math..

[4]  Volker Strehl.  Identities of Rothe-Abel- Schläfli-Hurwitz-type , 1992, Discret. Math..

[5]  E. Bender,et al.  Random mappings with constraints on coalescence and number of origins , 1982 .

[6]  N. H. Abel.  Beweis eines Ausdruckes, von welchem die Binomial-Formel ein einzelner Fall ist. , 1826 .

[7]  Ljuben Mutafčiev,et al.  Probability Distributions and Asymptotics for some Characteristics of Random Mappings , 1985 .

[8]  On some properties and applications of quasi-binomial distribution , 1990 .

[9]  A. Meir,et al.  The distance between points in random trees , 1970 .

[10]  Y. D. Burtin.  On a simple formula for random mappings and its applications , 1980, Journal of Applied Probability.

[11]  Abel series distributions with applications to fluctuations of sample functions of stochastic processes , 1990 .

[12]  J. Pitman.  The asymptotic behavior of the Hurwitz binomial distribution , 1997 .

[13]  A. Hurwitz,et al.  Über Abel's Verallgemeinerung der binomischen Formel , 1902 .

[14]  A. Joyal,et al.  Une th6orie combinatoire des s6ries formelles , 1981 .

[15]  Jim Pitman,et al.  Construction of markovian coalescents , 1998 .

[16]  George Bruce Halsted,et al.  The Collected Mathematical Papers of Arthur Cayley , 1899 .

[17]  Jerzy Jaworski.  A Random Bipartite Mapping , 1985 .

[18]  C. Dorea.  Connectivity of random graphs , 1982, Journal of Applied Probability.

[19]  A. J. Stam,et al.  Two identities in the theory of polynomials of binomial type , 1987 .

[20]  Jiang Zeng.  Multinomial convolution polynomials , 1996, Discret. Math..

[21]  J. Pitman.  Coalescent Random Forests , 1997 .

[22]  S. Ross.  A random graph , 1981 .

[23]  L. Mutafchiev,et al.  RANDOM MAPPINGS WITH AN ATTRACTING CENTER: LAGRANGIAN DISTRIBUTIONS AND A REGRESSION FUNCTION , 1990 .

[24]  Andrei Z. Broder,et al.  Generating random spanning trees , 1989, 30th Annual Symposium on Foundations of Computer Science.

[25]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[26]  Johannes Lenhard,et al.  Six Ways of Looking at Burtin's Lemma , 1999 .

[27]  Jerzy Jaworski.  On a random mapping (T, Pj ) , 1984 .

[28]  Walter J. Gutjahr.  Connection Reliabilities in Stochastic Acyclic Networks , 1994, Random Struct. Algorithms.

[29]  Andrei Z. Broder.  A GENERAL EXPRESSION FOR ABELIAN IDENTITIES , 1983 .

[30]  J. Jensen,et al.  Sur une identité d'Abel et sur d'autres formules analogues , 1902 .

[31]  David Aldous,et al.  Asymptotic Fringe Distributions for General Families of Random Trees , 1991 .

[32]  Krzysztof Nowicki,et al.  Statistical inference for a class of modified power series distributions with applications to random mapping theory , 1991 .

[33]  L. E. Clarke.  On Cayley's Formula for Counting Trees , 1958 .

[34]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[35]  P. C. Consul,et al.  A new urn model with predetermined strategy , 1975 .

[36]  V. E. Stepanov.  Random Mappings with a Single Attracting Center , 1971 .

[37]  L. Mutafchiev.  On random mappings with a single attracting centre , 1987, Journal of Applied Probability.

[38]  John Riordan,et al.  Inverse Relations and Combinatorial Identities , 1964 .

[39]  Jim Pitman.  The Multinomial Distribution on Rooted Labeled Forests , 1997 .

[40]  D. J. Aldous.  Brownian bridge asymptotics for random mappings , 1992, Advances in Applied Probability.

[41]  John Riordan,et al.  Enumeration of Linear Graphs for Mappings of Finite Sets , 1962 .

[42]  S. Berg,et al.  Modified binomial and poisson distributions with applications in random mapping theory , 1988 .