OFFSET
1,5
COMMENTS
Number of rises s_{i+1} > s_i in the RGS [s_1, ..., s_n] for a set partition of {1, ..., n}, where s_i is the index of the subset containing i, s_1 = 1 and s_i <= 1 + max_{j<i} s_j.
REFERENCES
W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000. [apparently unpublished, Joerg Arndt, Mar 05 2016]
LINKS
Alois P. Heinz, Rows n = 1..100, flattened
EXAMPLE
For example [1, 2, 1, 2, 2, 3] is the RGS for a set partition of {1, 2, 3, 4, 5, 6} and has 3 rises, at i = 1, i = 3 and i = 5.
1;
1,1;
1,3,1;
1,6,7,1;
1,10,26,14,1;
1,15,71,89,26,1;
1,21,161,380,267,46,1;
1,28,322,1268,1709,732,79,1;
1,36,588,3571,8136,6794,1887,133,1;
1,45,1002,8878,31532,44924,24717,4654,221,1;
1,55,1617,20053,104927,234412,221857,84170,11113,364,1;
1,66,2497,41965,310255,1025377,1528351,1006028,272557,25903,596,1;
MAPLE
b:= proc(n, i, m) option remember; expand(
`if`(n=0, x, add(b(n-1, 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] = Expand[If[n == 0, x, 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 23 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
AUTHOR
Winston C. Yang (winston(AT)cs.wisc.edu), Aug 31 2000
EXTENSIONS
More terms from Franklin T. Adams-Watters, Jun 08 2006
Clarified definition and edited comment and example, Joerg Arndt, Mar 05 2016
STATUS
approved