A057669 Triangle T(n,k) of number of minimal 3-covers of an unlabeled n+3-set that cover k points of that set uniquely (k=3,..,n+3).
1, 2, 1, 4, 3, 2, 7, 7, 6, 3, 11, 13, 14, 9, 4, 16, 22, 26, 21, 13, 5, 23, 34, 44, 40, 31, 17, 7, 31, 50, 68, 68, 59, 41, 23, 8, 41, 70, 100, 106, 101, 79, 55, 28, 10, 53, 95, 140, 157, 158, 136, 106, 68, 35, 12, 67, 125, 190, 221, 234, 214, 182, 132, 85, 42, 14, 83, 161
Offset: 0
Examples
[1], [2, 1], [4, 3, 2], [7, 7, 6, 3], ...
There are 7 minimal 3-covers of an unlabeled 6-set that cover 3 points of that set uniquely: {{1}, {2, 4, 5, 6}, {3, 4, 5, 6}}, {{1, 6}, {2, 4, 5}, {3, 4, 5, 6}}, {{1, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}}, {{1, 5, 6}, {2, 4, 6}, {3, 4, 5}}, {{1, 5, 6}, {2, 4, 6}, {3, 4, 5, 6}}, {{1, 5, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}}, {{1, 4, 5, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}}.
Links
- V. Jovovic, More information
Formula
T(n, k) = b(n, k)-b(n-1, k); b(n, k) = coefficient of x^k in x^3/6*(Z(S_n; 5+3*x, 5+3*x^2, ...)+3*Z(S_n; 3+x, 5+3*x^2, 3+x^3, 5+3*x^4, ...)+2*Z(S_n; 2, 2, 5+3*x^3, 2, 2, 5+3*x^6, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.
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