A255936 - OEIS

%I #34 Oct 23 2015 03:48:49

%S 0,0,1,95,23360,17853159,47300505935,455725535985152,

%T 16477833186525760257,2285218507961233452756479,

%U 1234874616385516438189472371200,2628743329824106687023439956782224783,22201933512060923158839975337648286975677119

%N Number of n X n binary matrices having a contiguous 2 by 2 submatrix whose every element is 1.

%C This sequence is of interest in the theory of 'Reliability in Engineering', where a 2-dimensional m by n array of elements might be considered to have failed if and only if it includes a contiguous r by s rectangle of failed elements. This is an extension of the 1-dimensional problem exemplified by a sequence of pumping stations along a pipeline. In the case where each element fails with probability 1/2, independently of the other elements, computing the probability that the system fails becomes a combinatorial problem.

%H S. R. Cowell, <a href="http://dx.doi.org/10.1155/2015/140909">A formula for the reliability of a d-dimensional consecutive-k-out-of-n:F system</a>, International Journal of Combinatorics, Vol. 2015, Article ID 140909, 5 pages

%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>

%F a(n) = A002416(n) - A139810(n). - _Alois P. Heinz_, Mar 11 2015

%F a(n) = -2^(n^2) Sum_{J a nonempty subset of E} (-1)^|J| Prod_{J' a nonempty subset of J} exp[(-1)^|J'| log(2) max(0, 2-(max_{e in J'} e_1 - min_{e in J'} e_1)) max(0, 2-(max_{e in J'} e_2 - min_{e in J'} e_2))], for n >= 2, where E={1,..,n-1} x {1,..,n-1}. (special case of Corollary 2 in the Cowell reference) - _Simon Cowell_, Sep 07 2015

%t failurenumber[m_, n_, r_, s_] :=

%t Module[{numberofnodes, numberofcases, cases, badsubmatrix,

%t    failurecases, i, j},

%t   numberofnodes = m n;

%t   numberofcases = 2^numberofnodes;

%t   cases = Tuples[{0, 1}, {m, n}];

%t   badsubmatrix = Table[1, {r}, {s}];

%t   failurecases =

%t    Parallelize[

%t     Select[cases,

%t      Apply[Or,

%t        Flatten[Table[#[[i ;; i + r - 1, j ;; j + s - 1]] ==

%t           badsubmatrix, {i, 1, m - r + 1}, {j, 1, n - s + 1}]]] &]];

%t   Length[failurecases] ]

%t failurenumberslist = Map[failurenumber[#1, #1, 2, 2] &, Range[2, 5]]

%Y Cf. A002416, A139810.

%K nonn

%O 0,4

%A _Simon Cowell_, Mar 11 2015

%E a(6)-a(12) from _Alois P. Heinz_, Mar 11 2015