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%I #9 May 10 2013 12:44:32
%S 2,18,133,873,5182,27786,135370,602454,2466628,9358497,33134431,
%T 110184932,346141949,1032550097,2938104492,8006865684,20971632456,
%U 52958252851,129291697111,305924724070,703108665327,1572722761341
%N Number of 5 x n binary matrices with 2 unit columns up to row and column permutations.
%C A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 5 - covers of an unlabeled n - set that cover 7 points of that set uniquely (if offset is 7).
%H Vladeta Jovovic, <a href="/A056885/a056885.pdf">Number of minimal covers of an unlabeled n - set that cover k points of that set uniquely</a>
%H Vladeta Jovovic, <a href="/A057972/a057972.pdf">Number of binary matrices with fixed number of unit columns up to row and column permutations</a>
%F Number of 5 x n binary matrices with k unit columns up to row and column permutations is coefficient of x^k in (1/5!)*(Z(S_n; 27 + 5*x, 27 + 5*x^2, ...) + 10*Z(S_n; 13 + 3*x, 27 + 5*x^2, 13 + 3*x^3, 27 + 5*x^4, ...) + 15*Z(S_n; 7 + x, 27 + 5*x^2, 7 + x^3, 27 + 5*x^4, ...) + 20*Z(S_n; 6 + 2*x, 6 + 2*x^2, 27 + 5*x^3, 6 + 2*x^4, 6 + 2*x^5, 27 + 5*x^6, ...) +
%F 20*Z(S_n; 4, 6 + 2*x^2, 13 + 3*x^3, 6 + 2*x^4, 4, 27 + 5*x^6, 4, 6 + 2*x^8, 13 + 3*x^9, 6 + 2*x^10, 4, 27 + 5*x^12, ...) + 30*Z(S_n; 3 + x, 7 + x^2, 3 + x^3, 27 + 5*x^4, 3 + x^5, 7 + x^6, 3 + x^7, 27 + 5*x^8, ...) + 24*Z(S_n; 2, 2, 2, 2, 27 + 5*x^5, 2, 2, 2, 2, 27 + 5*x^10, ...)),
%F where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.
%F G.f.: x^2/120*(15/(1 - x^1)^27 + 70/(1 - x^1)^13/(1 - x^2)^7 + 45/(1 - x^1)^7/(1 - x^2)^10 + 60/(1 - x^1)^6/(1 - x^3)^7 + 20/(1 - x^1)^4/(1 - x^2)^1/(1 - x^3)^3/(1 - x^6)^2 + 30/(1 - x^1)^3/(1 - x^2)^2/(1 - x^4)^5).
%Y Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963-A057968, A057970-A057972, A057969, A057970, A057972.
%K nonn
%O 2,1
%A _Vladeta Jovovic_, Oct 21 2000
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