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Increasing consecutive patterns in words

  • Mingjia YangEmail author
  • Doron Zeilberger
Article
  • 7 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

Primary 05A15 Secondary 05A05 05–04 

Notes

Acknowledgements

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.

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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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