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%I #11 Jul 14 2022 21:41:49
%S 1,5,24,115,551,2542,11193,46547,182164,670476,2325506,7624434,
%T 23716419,70253721,198905506,540079754,1410786483,3555443969,
%U 8667153126,20484365167,47037898503,105143200252,229178029000
%N 5 x n binary matrices without 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 5 points of that set uniquely (if offset is 5).
%H Vladeta Jovovic, <a href="/A056885/a056885.pdf">The 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 a(n)=(1/5!)*(Z(S_n; 27, 27, ...) + 10*Z(S_n; 13, 27, 13, 27, ...) + 15*Z(S_n; 7, 27, 7, 27, ...) + 20*Z(S_n; 6, 6, 27, 6, 6, 27, ...) + 20*Z(S_n; 4, 6, 13, 6, 4, 27, 4, 6, 13, 6, 4, 27, ...) + 30*Z(S_n; 3, 7, 3, 27, 3, 7, 3, 27, ...) + 24*Z(S_n; 2, 2, 2, 2, 27, 2, 2, 2, 2, 27, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.
%F G.f. : 1/120*(1/(1 - x^1)^27 + 10/(1 - x^1)^13/(1 - x^2)^7 + 15/(1 - x^1)^7/(1 - x^2)^10 + 20/(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 + 24/(1 - x^1)^2/(1 - x^5)^5).
%Y Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963-A057968, A057970-A057972.
%K nonn
%O 0,2
%A _Vladeta Jovovic_, Oct 20 2000
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