Pattern avoidance in matrices

Sergey Kitaev, Toufik Mansour, Antoine Vella

Research output: Contribution to journalArticlepeer-review


We generalize the concept of pattern avoidance from words to matrices, and consider specifically binary matrices avoiding the smallest non-trivial patterns. For all binary right angled patterns (0/1 subconfigurations with 3 entries, 2 in the same row and 2 in the same column) and all 2 × 2 binary patterns, we enumerate the m × n binary matrices avoiding the given pattern. For right angled patterns, and the all zeroes 2 × 2 pattern, we employ direct combinatorial considerations to obtain either explicit closed form formulas or generating functions; in the other cases, we use the transfer matrix method to derive an algorithm which gives, for any fixed m, a closed form formula in n. Some of these cases lead naturally to extremal problems of Ramsey type.

Original languageEnglish
JournalJournal of Integer Sequences
Issue number2
StatePublished - 7 Apr 2005


  • Binary matrices
  • Forbidden submatrices
  • Forbidden subsequences
  • Ramsey Theory
  • Transfer matrix

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


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