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A General Asymptotic Scheme for the Analysis of Partition Statistics

Published online by Cambridge University Press:  08 September 2014

PETER J. GRABNER ,
ARNOLD KNOPFMACHER  and
STEPHAN WAGNER
PETER J. GRABNER
Affiliation:
Institut für Analysis und Computational Number Theory, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria (e-mail: peter.grabner@tugraz.at)
ARNOLD KNOPFMACHER
Affiliation:
The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, PO Wits, 2050 Johannesburg, South Africa (e-mail: arnold.knopfmacher@wits.ac.za)
STEPHAN WAGNER
Affiliation:
Department of Mathematical Sciences, Stellenbosch University, 7602 Stellenbosch, South Africa (e-mail: swagner@sun.ac.za)
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Abstract

We consider statistical properties of random integer partitions. In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P(x)F(x), where P(x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtained in a quasi-automatic way from expansions of F(x) around x = 1, which parallels the classical singularity analysis of Flajolet and Odlyzko in many ways. Numerous examples from the literature, as well as some new statistics, are treated via this methodology. In addition, we show how to compute further terms in the asymptotic expansions of previously studied partition statistics.

Keywords

Primary 11P82Secondary 05A1605A17
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 6: Honouring the Memory of Philippe Flajolet - Part 2 , November 2014 , pp. 1057 - 1086
Copyright
Copyright © Cambridge University Press 2014 

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