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The Distribution of Ascents of Size d or More in Partitions of n

Published online by Cambridge University Press:  01 July 2008

CHARLOTTE BRENNAN ,
ARNOLD KNOPFMACHER  and
STEPHAN WAGNER
CHARLOTTE BRENNAN
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa (e-mail: [email protected])
ARNOLD KNOPFMACHER
Affiliation:
The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa (e-mail: [email protected])
STEPHAN WAGNER
Affiliation:
Institute for Analysis and Computational Number Theory, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria (e-mail: [email protected])
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Abstract

A partition of a positive integer n is a finite sequence of positive integers a1, a2, . . ., ak such that a1+a2+ċ ċ ċ+ak=n and ai+1 ≥ ai for all i. Let d be a fixed positive integer. We say that we have an ascent of size d or more if ai+1 ≥ ai+d.

We determine the mean, the variance and the limiting distribution of the number of ascents of size d or more (equivalently, the number of distinct part sizes of multiplicity d or more) in the partitions of n.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 4 , July 2008 , pp. 495 - 509
Copyright
© Cambridge University Press 2008

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