Omnibus Sequences, Coupon Collection, and Missing Word Counts

  • Sunil Abraham
  • Greg Brockman
  • Stephanie Sapp
  • Anant P. Godbole


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 \({\mathbb E}(M)\).


Coupon collection Omnibus sequences Extreme value distribution 

AMS 2000 Subject Classification



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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Sunil Abraham
    • 1
  • Greg Brockman
    • 2
  • Stephanie Sapp
    • 3
  • Anant P. Godbole
    • 4
  1. 1.Oxford UniversityOxfordUK
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA
  3. 3.University of CaliforniaBerkeleyUSA
  4. 4.East Tennessee State UniversityJohnson CityUSA

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