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Successions in Words and Compositions

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Abstract

We consider words over the alphabet [k] = {1, 2, . . . , k}, k ≥ 2. For a fixed nonnegative integer p, a p-succession in a word w 1 w 2 . . . w n consists of two consecutive letters of the form (w i , w i  + p), i = 1, 2, . . . , n − 1. We analyze words with respect to a given number of contained p-successions. First we find the mean and variance of the number of p-successions. We then determine the distribution of the number of p-successions in words of length n as n (and possibly k) tends to infinity; a simple instance of a phase transition (Gaussian-Poisson-degenerate) is encountered. Finally, we also investigate successions in compositions of integers.

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Authors and Affiliations

  1. The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg, South Africa

    Arnold Knopfmacher & Augustine Munagi

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

    Stephan Wagner

Authors
  1. Arnold Knopfmacher
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  2. Augustine Munagi
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  3. Stephan Wagner
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Correspondence to Arnold Knopfmacher.

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Knopfmacher, A., Munagi, A. & Wagner, S. Successions in Words and Compositions. Ann. Comb. 16, 277–287 (2012). https://doi.org/10.1007/s00026-012-0131-z

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  • DOI: https://doi.org/10.1007/s00026-012-0131-z

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