

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
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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. AdamsWatters, 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 [Apparently unpublished]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(n1, j, max(m, j))*
`if`(j<i, x, 1), j=1..m+1)))
end:
T:= n> (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 (* JeanFrançois Alcover, May 24 2016, after Alois P. Heinz *)


CROSSREFS

Cf. A000110 (row sums).
Cf. A056857A056863.
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. AdamsWatters, Jun 08 2006


STATUS

approved



