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Omnibus Sequences, Coupon Collection, and Missing Word Counts

Author

Listed:
  • Sunil Abraham

    (Oxford University)

  • Greg Brockman

    (Massachusetts Institute of Technology)

  • Stephanie Sapp

    (University of California)

  • Anant P. Godbole

    (East Tennessee State University)

Abstract

In this paper, we study the properties of k-omnisequences of length n, defined to be strings of length n that contain all strings of smaller length k embedded as (not necessarily contiguous) subsequences. We start by proving an elementary result that relates our problem to the classical coupon collector problem. After a short survey of relevant results in coupon collection, we focus our attention on the number M of strings (or words) of length k that are not found as subsequences of an n string, showing that there is a gap between the probability threshold for the emergence of an omnisequence and the zero-infinity threshold for ${\mathbb E}(M)$ .

Suggested Citation

  • Sunil Abraham & Greg Brockman & Stephanie Sapp & Anant P. Godbole, 2013. "Omnibus Sequences, Coupon Collection, and Missing Word Counts," Methodology and Computing in Applied Probability, Springer, vol. 15(2), pages 363-378, June.
    • Handle: RePEc:spr:metcap:v:15:y:2013:i:2:d:10.1007_s11009-011-9247-6
    DOI: 10.1007/s11009-011-9247-6
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    Cited by:

    1. Anant P. Godbole & Martha Liendo, 2016. "Waiting Time Distribution for the Emergence of Superpatterns," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 517-528, June.
    2. Angel, Omer & Matzavinos, Anastasios & Roitershtein, Alexander, 2019. "Limit theorem for the Robin Hood game," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 9-15.

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