A139810 -id:A139810 - cp's OEIS frontend

A255936 Number of n X n binary matrices having a contiguous 2 by 2 submatrix whose every element is 1.

0, 0, 1, 95, 23360, 17853159, 47300505935, 455725535985152, 16477833186525760257, 2285218507961233452756479, 1234874616385516438189472371200, 2628743329824106687023439956782224783, 22201933512060923158839975337648286975677119

Offset: 0

Simon Cowell, Mar 11 2015

nonn

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.

S. R. Cowell, A formula for the reliability of a d-dimensional consecutive-k-out-of-n:F system, International Journal of Combinatorics, Vol. 2015, Article ID 140909, 5 pages
Index entries for sequences related to binary matrices

Cf. A002416, A139810.

failurenumber[m_, n_, r_, s_] :=
Module[{numberofnodes, numberofcases, cases, badsubmatrix,
   failurecases, i, j},
  numberofnodes = m n;
  numberofcases = 2^numberofnodes;
  cases = Tuples[{0, 1}, {m, n}];
  badsubmatrix = Table[1, {r}, {s}];
  failurecases =
   Parallelize[
    Select[cases,
     Apply[Or,
       Flatten[Table[#[[i ;; i + r - 1, j ;; j + s - 1]] ==
          badsubmatrix, {i, 1, m - r + 1}, {j, 1, n - s + 1}]]] &]];
  Length[failurecases] ]
failurenumberslist = Map[failurenumber[#1, #1, 2, 2] &, Range[2, 5]]

a(n) = A002416(n) - A139810(n). - Alois P. Heinz, Mar 11 2015

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

a(6)-a(12) from Alois P. Heinz, Mar 11 2015

cp's OEIS Frontend

A255936 Number of n X n binary matrices having a contiguous 2 by 2 submatrix whose every element is 1.

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