OFFSET
0,4
COMMENTS
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.
LINKS
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
FORMULA
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
MATHEMATICA
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]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Simon Cowell, Mar 11 2015
EXTENSIONS
a(6)-a(12) from Alois P. Heinz, Mar 11 2015
STATUS
approved