The distribution of ascents of size d or more in partitions of n

  • Authors:

  • Charlotte Brennan;Arnold Knopfmacher;Stephan Wagner

  • Affiliations:

  • School of mathematics, university of the witwatersrand, private bag 3, wits 2050, johannesburg, south africa (e-mail: charlotte.brennan@wits.ac.za);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: arnold. ...;Institute for analysis and computational number theory, graz university of technology, steyrergasse 30, 8010 graz, austria (e-mail: wagner@finanz.math.tugraz.at)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2008

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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.