Geodesic
Pattern avoidance in matrices
Journal of integer sequences, Tome 8 (2005) no. 2
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.
Classification : 05A05, 05A15, 05D10, 05C50
Keywords: binary matrices, Ramsey theory, forbidden subsequences, forbidden submatrices, transfer matrix 
@article{JIS_2005__8_2_a4,
     author = {Kitaev,  Sergey and Mansour,  Toufik and Vella,  Antoine},
     title = {Pattern avoidance in matrices},
     journal = {Journal of integer sequences},
     year = {2005},
     volume = {8},
     number = {2},
     zbl = {1064.05015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2005__8_2_a4/}
}
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Kitaev,  Sergey; Mansour,  Toufik; Vella,  Antoine. Pattern avoidance in matrices. Journal of integer sequences, Tome 8 (2005) no. 2. http://geodesic.mathdoc.fr/item/JIS_2005__8_2_a4/