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Search: id:A056861
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| A056861 |
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Triangle T(n,k) = number of set partitions of {1..n} that have an increase at index k (1<=k<n). |
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2
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| 1, 3, 2, 10, 7, 6, 37, 27, 23, 21, 151, 114, 97, 88, 83, 674, 523, 446, 403, 378, 363, 3263, 25, 89, 2217, 1999, 1867, 1785, 1733, 17007, 13744, 11829, 10658, 9923, 9452, 9145, 8942, 9482, 8, 77821, 67340, 60689, 56380, 53541, 51644, 50361, 49484, 562595
(list; table; graph; listen)
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OFFSET
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2,2
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COMMENT
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Number of rises s_{k+1} > s_k in a set partition {s_1, ..., s_n} of {1, ..., n}, where s_i is the subset containing i, s(1) = 1, and s(i) <= 1 + max of previous s(j)'s.
Note that the number of equalities at any index is B(n-1), where B(n) are the Bell numbers. Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 08 2006
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REFERENCES
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W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000.
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EXAMPLE
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For example {1, 2, 1, 2, 2, 3} is a set partition of {1, 2,
3, 4, 5, 6} and has 3 rises, at i = 1, i = 3, and i = 5.
1; 3,2; 10,7,6; 37,27,23,21; 151,114,97,88,83; ...
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CROSSREFS
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Cf. Bell numbers A000110.
Cf. A056857-A056863.
Sequence in context: A063549 A057977 A071653 this_sequence A103245 A019242 A064367
Adjacent sequences: A056858 A056859 A056860 this_sequence A056862 A056863 A056864
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Winston C. Yang (winston(AT)cs.wisc.edu), Aug 31 2000
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EXTENSIONS
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Edited and extended by Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 08 2006 Franklin T. Adams-Watters
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