Pattern avoidance in matrices

Sergey Kitaev, Toufik Mansour, Antoine Vella

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


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 x 2 binary patterns, we enumerate the m x n binary matrices avoiding the given pattern. For right angled patterns, and the all zeroes 2 x 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
Article number05.2.2
Number of pages16
JournalJournal of Integer Sequences
Publication statusPublished - 2005


  • binary matrices
  • transfer matrix
  • forbidden submatrices
  • Ramsey theory
  • forbidden subsequences


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