Increasing consecutive patterns in words

  • Mingjia YangEmail author
  • Doron Zeilberger


We show how to enumerate words in \(1^{m_1} \ldots n^{m_n}\) that avoid the increasing consecutive pattern \(12 \ldots r\) for any \(r \ge 2\). Our approach yields an \(O(n^{s+1})\) algorithm to enumerate words in \(1^s \ldots n^s\), avoiding the consecutive pattern \(1\ldots r\), for any s, and any r. This enables us to supply many more terms to quite a few OEIS sequences and create new ones. We also treat the more general case of counting words with a specified number of the pattern of interest (the avoiding case corresponding to zero appearances). This article is accompanied by three Maple packages implementing our algorithms.


Permutation Word Consecutive pattern Generating function Efficient computation Goulden–Jackson cluster method 

Mathematics Subject Classification

Primary 05A15 Secondary 05A05 05–04 



Many thanks are due to Sergi Elizalde for help with the references, to Yonah Biers-Ariel for suggestions on the format of the paper, and to Justin Troyka for pointing out that “our” Theorem 1 appeared in Ira Gessel’s Ph.D. thesis. Also special thanks are due to the anonymous referee for the helpful suggestions.


  1. 1.
    Baxter, A., Nakamura, B., Zeilberger, D.: Automatic generation of theorems and proofs on enumerating consecutive-Wilf classes. In: Kotsireas, I., Zima, E. (eds.) Advances in Combinatorics: Waterloo Workshop in Computer Algebra, W80, May 26–29, 2011 [Volume in Honor of Herbert S. Wilf], pp. 121–138. Springer, New York (2013)CrossRefGoogle Scholar
  2. 2.
    Burstein, A.: Enumeration of words with forbidden patterns. Ph.D. thesis, University of Pennsylvania (1998)Google Scholar
  3. 3.
    Burstein, A., Mansour, T.: Words restricted by patterns with at most 2 distinct letters. Electron. J. Combin. 9, #R3 (2002)Google Scholar
  4. 4.
    Davis, F.N., Barton, D.E.: Combinatorial Chance. Hafner, New York (1962)Google Scholar
  5. 5.
    Elizalde, S., Noy, M.: Consecutive patterns in permutations. Adv. Appl. Math. 30, 110–125 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gessel, I.M.: Generating functions and enumeration of sequences. Ph.D. thesis, Massachusetts Institute of Technology (1977)Google Scholar
  7. 7.
    Gessel, I.M.: Symmetric functions and P-recursiveness. J. Combin. Theory Ser. A 53, 257–285 (1990)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Goulden, I., Jackson, D.: An inversion theorem for cluster decomposition of sequences with distinguished subsequences. J. Lond. Math. Soc. (2) 20, 567–576 (1979)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Mendes, A., Remmel, J.B.: Permutations and words counted by consecutive patterns. Adv. Appl. Math. 37, 443–480 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Nakamura, B.: Computational approaches to consecutive pattern avoidance in permutations. Pure Math. Appl. (PU.M.A.) 22, 253–268 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Noonan, J., Zeilberger, D.: The Goulden–Jackson cluster method: extensions, applications, and implementations. J. Difference Equ. Appl. 5, 355–377 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Pudwell, L.: Enumeration schemes for pattern-avoiding words and permutations. Ph.D. thesis, Rutgers University (2008). Accessed 28 Sept 2018
  13. 13.
    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences.
  14. 14.
    Wen, X.: The symbolic Goulden–Jackson cluster method. J. Difference Equ. Appl. 11, 173–179 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsRutgers University (New Brunswick)PiscatawayUSA

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