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 A056859 Triangle of number of falls in set partitions of n. 1
 1, 2, 0, 4, 1, 0, 8, 7, 0, 0, 16, 32, 4, 0, 0, 32, 121, 49, 1, 0, 0, 64, 411, 360, 42, 0, 0, 0, 128, 1304, 2062, 624, 22, 0, 0, 0, 256, 3949, 10163, 6042, 730, 7, 0, 0, 0, 512, 11567, 45298, 45810, 12170, 617, 1, 0, 0, 0, 1024, 33056, 187941, 296017, 141822, 18325, 385, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Number of falls s_i > s_{i+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. The maximum number of falls is in a set partition like 1,2,1,3,2,1,... - Franklin T. Adams-Watters, Jun 08 2006 REFERENCES W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000. [Apparently unpublished] LINKS Alois P. Heinz, Rows n = 1..100, flattened EXAMPLE For example {1, 2, 1, 2, 2, 3} is a set partition of {1, 2, 3, 4, 5, 6} and has 1 fall, at i = 2. T(n=3,f=0)=4 counts the partitions {1,1,1}, {1,1,2}, {1,2,2}, and {1,2,3}. T(n=3,f=1) counts the partition {1,2,1}. - R. J. Mathar, Mar 04 2016 1; 2,0; 4,1,0; 8,7,0,0; 16,32,4,0,0; 32,121,49,1,0,0; 64,411,360,42,0,0,0; 128,1304,2062,624,22,0,0,0; 256,3949,10163,6042,730,7,0,0,0; 512,11567,45298,45810,12170,617,1,0,0,0; 1024,33056,187941,296017,141822,18325,385,0,0,0,0; 2048,92721,739352,1708893,1318395,330407,21605,176,0,0,0,0; MAPLE b:= proc(n, i, m) option remember;       `if`(n=0, x, expand(add(b(n-1, j, max(m, j))*       `if`(j (p-> seq(coeff(p, x, i), i=1..n))(b(n, 1, 0)): seq(T(n), n=1..12);  # Alois P. Heinz, Mar 24 2016 MATHEMATICA b[n_, i_, m_] := b[n, i, m] = If[n == 0, x, Expand[Sum[b[n - 1, j, Max[m, j]]*If[j < i, x, 1], {j, 1, m + 1}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 1, 0]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, May 24 2016, after Alois P. Heinz *) CROSSREFS Cf. A000110 (row sums). Cf. A056857-A056863. Sequence in context: A140648 A153342 A144258 * A272098 A291929 A327807 Adjacent sequences:  A056856 A056857 A056858 * A056860 A056861 A056862 KEYWORD easy,nonn,tabl AUTHOR Winston C. Yang (winston(AT)cs.wisc.edu), Aug 31 2000 EXTENSIONS Corrected and extended by Franklin T. Adams-Watters, Jun 08 2006 STATUS approved

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Last modified January 21 13:55 EST 2020. Contains 331113 sequences. (Running on oeis4.)