On the Exact Distributions of Pattern Statistics for a Sequence of Binary Trials: A Combinatorial Approach

  • Frosso S. MakriEmail author
  • Zaharias M. Psillakis
Living reference work entry


Consider a sequence of exchangeable or Markov-dependent binary (zero-one) trials. A sequence of independent and identically distributed binary trials is covered as a particular case of both the prementioned ones. For counting/waiting time pattern statistics defined on such model sequences, we point out how their exact probability distributions can be established using enumerative combinatorics. The expressions for the distributions contain probabilities depending on the internal structure of the model sequence and combinatorial numbers denoting set cardinalities. The latter numbers depend on the considered pattern statistics and the number of ones, for an exchangeable sequence, as well as the number of runs of ones, for a Markov-dependent sequence. These numbers become concrete when certain patterns and enumerative schemes are studied on the model sequences. Exact distributions for statistics connected to patterns of limited length, as well as to certain runs and scans, are provided using proper combinatorial numbers and exemplify the approach.


Exact distributions Enumerative combinatorics Runs, scans, and patterns Binary trials 



The authors wish to thank the anonymous referee for the thorough reading and useful comments and suggestions which helped to improve the paper.


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Copyright information

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

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PatrasPatrasGreece
  2. 2.Department of PhysicsUniversity of PatrasPatrasGreece

Section editors and affiliations

  • Joseph Glaz
    • 1
  • Markos V. Koutras
    • 2
  1. 1.Department of StatisticsUniversity of ConnecticutStorrsUSA
  2. 2.Department of Statistics and Insurance ScienceUniversity of PiraeusPiraeusGreece

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