##
**
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.

**
Full version: pdf,
dvi,
ps,
latex
**

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

Return to
**Journal of Integer Sequences home page**