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Compositions with a fixed number of inversions

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Abstract

A composition of the positive integer n is a representation of n as an ordered sum of positive integers \(n=a_1+a_2+\dots +a_m.\) There are \(2^{n-1}\) unrestricted compositions of n, which can be sorted according to the number of inversions they contain. (An inversion in a composition is a pair of summands \(\{a_i, a_j\}\) for which \( i< j\) and \(a_i>a_j\).) The number of inversions of a composition is an indication of how far the composition is from a partition of n, which by convention uses a sequence of nondecreasing summands and thus has no inversions. We count compositions of n with exactly r inversions in several ways to derive generating function identities, and also consider asymptotic results.

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Authors and Affiliations

  1. The John Knopfmacher Center for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg, South Africa

    A. Knopfmacher

  2. West Virginia University, Morgantown, WV, 26506-6310, USA

    M. E. Mays

  3. Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, Matieland, 7602, South Africa

    S. Wagner

Authors
  1. A. Knopfmacher
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  2. M. E. Mays
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  3. S. Wagner
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Corresponding author

Correspondence to A. Knopfmacher.

Additional information

This material is based upon work supported by the National Research Foundation under Grant Numbers 2053740 and 96236. The second and third author thank the John Knopfmacher Centre for Applicable Analysis and Number Theory at The University of the Witwatersrand.

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Knopfmacher, A., Mays, M.E. & Wagner, S. Compositions with a fixed number of inversions. Aequat. Math. 93, 601–617 (2019). https://doi.org/10.1007/s00010-018-0563-6

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  • DOI: https://doi.org/10.1007/s00010-018-0563-6

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