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A058710
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Triangle T(n,k) giving number of loopless matroids of rank k on n labeled points (n >= 0, 0 <= k <= n).
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8
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1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 14, 11, 1, 0, 1, 51, 106, 26, 1, 0, 1, 202, 1232, 642, 57, 1, 0, 1, 876, 22172, 28367, 3592, 120, 1, 0, 1, 4139, 803583, 8274374, 991829, 19903, 247, 1
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OFFSET
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0,9
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COMMENTS
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The old references have some typos, some of which were corrected in the recent references (in 2004). Few additional typos were corrected here from the recent references. Here are some of the changes: T(5,2) = 31 --> 51 (see the comment by Ralf Stephan below); T(5,4) = 21 --> 26; sum of row n=5 is 185 (not 160 or 165); T(8,3) = 686515 --> 803583; T(8, 6) = 19904 --> 19903, and some others.
This triangular array is the same as A058711 except that the current one has row n = 0 and column k = 0.
(End)
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LINKS
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FORMULA
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T(n,0) = 0^n for n >= 0.
T(n,1) = 1 for n >= 1.
T(n,3) = Sum_{i = 3..n} Stirling2(n,i) * (A056642(i) - 1) = Sum_{i = 3..n} A008277(n,i) * A058720(i,3) for n >= 3.
T(n,k) = Sum_{i = k..n} Stirling2(n,i) * A058720(i,k) for n >= k. [Dukes (2004), p. 3; see the equation with the Stirling numbers of the second kind.]
(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;
0, 1;
0, 1, 1;
0, 1, 4, 1;
0, 1, 14, 11, 1;
0, 1, 51, 106, 26, 1;
0, 1, 202, 1232, 642, 57, 1;
0, 1, 876, 22172, 28367, 3592, 120, 1;
0, 1, 4139, 803583, 8274374, 991829, 19903, 247, 1;
...
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CROSSREFS
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Cf. Same as A058711 (except for row n=0 and column k=0).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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