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Part Sizes of Smooth Supercritical Compositional Structures

Published online by Cambridge University Press:  09 July 2014

EDWARD A. BENDER  and
ZHICHENG GAO
EDWARD A. BENDER
Affiliation:
Department of Mathematics, UCSD, La Jolla, CA 92093-0112, USA (e-mail: ebender@ucsd.edu)
ZHICHENG GAO
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S5B6, Canada (e-mail: zgao@math.carleton.ca)
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Abstract

We define the notion of smooth supercritical compositional structures. Two well-known examples are compositions and graphs of given genus. The ‘parts’ of a graph are the subgraphs that are maximal trees. We show that large part sizes have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. In many cases this leads to asymptotic formulas for the probability of being gap-free and for the expected values of the largest part sizes, number of distinct parts and number of parts of multiplicity k.

Keywords

Primary 05A16Secondary 05A1505A0505C0505C1005C30

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 5: Honouring the Memory of Philippe Flajolet - Part 1 , September 2014 , pp. 686 - 716
Copyright
Copyright © Cambridge University Press 2014 

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