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Triangle read by rows: T(n,m) is the number of n X m binary (0,1) matrices that represent perfect parity patterns.
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%I #41 Nov 30 2014 14:57:05

%S 0,1,0,0,2,0,0,0,0,15,0,0,4,0,1,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,

%T 63,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,112,0,0,0,

%U 36,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0

%N Triangle read by rows: T(n,m) is the number of n X m binary (0,1) matrices that represent perfect parity patterns.

%C An n X m matrix of zeros and ones is perfect if no row or column consists entirely of zeros (as counted in A183109). It is a parity pattern if every 0 is adjacent (vertically or horizontally) to an even number of 1s and every 1 is adjacent to an odd number of 1s.

%H R. J. Mathar, <a href="/A225783/b225783.txt">Table of n, a(n) for n = 1..106</a>

%H R. Chapman, D. E. Knuth, <a href="http://www.jstor.org/stable/27642574">Problem 11243, Perfect parity patterns</a>, Am. Math. Monthly 115 (7) (2008) p 668.

%H R. J. Mathar, <a href="/A225783/a225783.pdf">Discussion and JAVA source code</a>

%e The T(5,3) = 4 perfect parity 5 X 3 patterns are

%e 0 0 1

%e 0 1 1

%e 1 0 1

%e 1 1 0

%e 1 0 0

%e ------

%e 0 1 1

%e 1 0 0

%e 1 0 1

%e 0 0 1

%e 1 1 0

%e --------

%e 1 0 0

%e 1 1 0

%e 1 0 1

%e 0 1 1

%e 0 0 1

%e --------

%e 1 1 0

%e 0 0 1

%e 1 0 1

%e 1 0 0

%e 0 1 1

%K nonn,tabl

%O 1,5

%A _R. J. Mathar_, Jun 13 2014