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A058669
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Triangle T(n,k) read by rows, giving number of matroids of rank k on n labeled points (n >= 0, 0 <= k <= n).
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5
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1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 36, 15, 1, 1, 31, 171, 171, 31, 1, 1, 63, 813, 2053, 813, 63, 1, 1, 127, 4012, 33442, 33442, 4012, 127, 1, 1, 255, 20891, 1022217, 8520812, 1022217, 20891, 255, 1, 1, 511
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OFFSET
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0,5
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LINKS
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FORMULA
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T(n,0) = 1 for n >= 0.
T(n,1) = 2^n - 1 for n >= 1. [Dukes (2004), Theorem 2.1 (ii).
T(n,2) = Bell(n+1) - 2^n = A000110(n+1) - A000079(n) for n >= 2. [Dukes (2004), Theorem 2.1 (ii).]
T(n,k) = Sum_{m = k..n} binomial(n,m) * A058711(m,k) for n >= k. [Dukes (2004), see the equations before Theorem 2.1.]
(End)
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EXAMPLE
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Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
1;
1, 1;
1, 3, 1;
1, 7, 7, 1;
1, 15, 36, 15, 1;
1, 31, 171, 171, 31, 1;
1, 63, 813, 2053, 813, 63, 1;
1, 127, 4012, 33442, 33442, 4012, 127, 1;
1, 255, 20891, 1022217, 8520812, 1022217, 20891, 255, 1;
...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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