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Search: id:A056862
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| A056862 |
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Triangle T(n,k) = number of set partitions of {1..n} that have a decrease at index k (1<=k<n). |
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2
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| 0, 0, 1, 0, 3, 4, 0, 10, 14, 16, 0, 37, 54, 63, 68, 0, 151, 228, 271, 296, 311, 0, 674, 1046, 126, 4, 1396, 1478, 1530, 0, 3263, 5178, 6349, 7084, 7555, 7862, 8065, 0, 17007, 27488, 34139, 38448, 41287, 43184, 44467, 45344, 0, 94828, 155642, 195494, 222044
(list; table; graph; listen)
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OFFSET
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2,5
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COMMENT
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Number of falls s_k > s_{k+1} 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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FORMULA
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T(n,k) = B(n) - B(n-1) - A056861(n,k) Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 08 2006
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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 1 fall, at i = 2.
0; 0,1; 0,3,4; 0,10,14,16; 0,37,54,63,68; ...
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CROSSREFS
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Cf. Bell numbers A000110.
Cf. A056857-A056863.
Adjacent sequences: A056859 A056860 A056861 this_sequence A056863 A056864 A056865
Sequence in context: A092894 A011338 A049251 this_sequence A113035 A099447 A078067
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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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