Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.2

Pattern Avoidance in Matrices

Sergey Kitaev
Department of Mathematics
University of Kentucky
Lexington, KY 40506-0027
USA

Toufik Mansour
Department of Mathematics
University of Haifa
31905 Haifa
Israel

Antoine Vella
Department of Combinatorics and Optimization
University of Waterloo
Waterloo, Ontario N2L 3G1
Canada

Abstract: 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.

Received December 8 2004; revised version received April 7 2005. Published in Journal of Integer Sequences April 7 2005.