%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