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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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References

[1]Andrews, G. E., Askey, R. and Roy, R. (1999) Special Functions, Cambridge University Press.CrossRefGoogle Scholar
[2]Apostol, T. M. (1990)Modular Functions and Dirichlet Series in Number Theory, 2nd edn, Vol. 41 of Graduate Texts in Mathematics, Springer, New York.Google Scholar
[3]Corteel, S., Pittel, B., Savage, C. and Wilf, H. (1999) On the multiplicity of parts in a random partition. Random Struct. Alg. 14 185–197.3.0.CO;2-F>CrossRefGoogle Scholar
[4]Curtiss, J. (1942) A note on the theory of moment generating functions. Ann. Math. Statist. 13 430–433.CrossRefGoogle Scholar
[5]Erdo″s, P. and Lehner, J. (1941) The distribution of the number of summands in the partitions of a positive integer. Duke Math. J. 8 335–345.Google Scholar
[6]Hwang, H.-K. (2001) Limit theorems for the number of summands in integer partitions. J. Combin. Theory Ser. A 96 89–126.CrossRefGoogle Scholar
[7]Knopfmacher, A. and Warlimont, R. (2006) Gaps in integer partitions. Utilitas Mathematica 71 257–267.Google Scholar
[8]Lewin, L. (1981) Polylogarithms and Associated Functions, North-Holland, New York.Google Scholar
[9]Wilf, H. (1983) Three problems in combinatorial asymptotics. J. Combin. Theory Ser. A 35 199–207.CrossRefGoogle Scholar