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FORMULA
| 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/120*(5/(1 - x^1)^27 + 30/(1 - x^1)^13/(1 - x^2)^7 + 15/(1 - x^1)^7/(1 - x^2)^10 + 40/(1 - x^1)^6/(1 - x^3)^7 + 30/(1 - x^1)^3/(1 - x^2)^2/(1 - x^4)^5).
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