3,1

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 8 points of that set uniquely (if offset is 8).

V. Jovovic, Number of minimal covers of an unlabeled n - set that cover k points of that set uniquely

V. Jovovic, Number of binary matrices with fixed number of unit columns up to row and column permutations

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, ...) + 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, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n. G.f. : x^3/120*(35/(1 - x^1)^27 + 130/(1 - x^1)^13/(1 - x^2)^7 + 45/(1 - x^1)^7/(1 - x^2)^10 + 100/(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).

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963 - A057968, A057969 - A057971.

Sequence in context: A121147 A114654 A111400 this_sequence A114849 A112425 A144579

Adjacent sequences: A057969 A057970 A057971 this_sequence A057973 A057974 A057975

nonn

Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 21 2000