OFFSET
2,5
COMMENTS
Number of falls s_k > s_{k+1} in a RGS [s_1, ..., s_n] of a set partition of {1, ..., n}, where s_i is the subset containing i, s_1 = 1 and s_i <= 1 + max(j<i, s_j).
Note that the number of equalities at any index is B(n-1), where B(n) are the Bell numbers. - 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 [apparently unpublished, Joerg Arndt, Mar 05 2016]
LINKS
Alois P. Heinz, Rows n = 2..100, flattened
FORMULA
T(n,k) = B(n) - B(n-1) - A056861(n,k). - Franklin T. Adams-Watters, Jun 08 2006
Conjecture: T(n,3) = 2*A011965(n). - R. J. Mathar, Mar 08 2016
EXAMPLE
For example, [1, 2, 1, 2, 2, 3] is the RGS of 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;
0,151,228,271,296,311;
0,674,1046,1264,1396,1478,1530;
0,3263,5178,6349,7084,7555,7862,8065;
0,17007,27488,34139,38448,41287,43184,44467,45344;
0,94828,155642,195494,222044,239976,252230,260690,266584,270724;
0,562595,935534,1186845,1358452,1476959,1559602,1617737,1658952,1688379, 1709526;
MAPLE
b:= proc(n, i, m, t) option remember; `if`(n=0, [1, 0],
add((p-> p+[0, `if`(j<i, p[1]*x^t, 0)])(
b(n-1, j, max(m, j), t+1)), j=1..m+1))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n-1))(b(n, 1, 0$2)[2]):
seq(T(n), n=2..12); # Alois P. Heinz, Mar 24 2016
MATHEMATICA
b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, {1, 0}, Sum[Function[p, p + {0, If[j<i, p[[1]]*x^t, 0]}][b[n-1, j, Max[m, j], t+1]], {j, 1, m+1}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n - 1}]][b[n, 1, 0, 0][[2]]];
Table[T[n], {n, 2, 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
Edited and extended by Franklin T. Adams-Watters, Jun 08 2006
Clarified definition and edited comment and example, Joerg Arndt, Mar 08 2016
Data corrected, R. J. Mathar, Mar 08 2016
STATUS
approved