A225783 Triangle read by rows: T(n,m) is the number of n X m binary (0,1) matrices that represent perfect parity patterns.

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, 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, 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

Offset: 1

Author

R. J. Mathar, Jun 13 2014

Keywords

• nonn
• tabl

Comments

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.

Examples

			The T(5,3) = 4 perfect parity 5 X 3 patterns are
0 0 1
0 1 1
1 0 1
1 1 0
1 0 0
------
0 1 1
1 0 0
1 0 1
0 0 1
1 1 0
--------
1 0 0
1 1 0
1 0 1
0 1 1
0 0 1
--------
1 1 0
0 0 1
1 0 1
1 0 0
0 1 1
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..106
• R. Chapman, D. E. Knuth, Problem 11243, Perfect parity patterns, Am. Math. Monthly 115 (7) (2008) p 668.
• R. J. Mathar, Discussion and JAVA source code