A056861 Triangle T(n,k) is the number of restricted growth strings (RGS) of set partitions of {1..n} that have an increase at index k (1<=k<n).

1, 3, 2, 10, 7, 6, 37, 27, 23, 21, 151, 114, 97, 88, 83, 674, 523, 446, 403, 378, 363, 3263, 2589, 2217, 1999, 1867, 1785, 1733, 17007, 13744, 11829, 10658, 9923, 9452, 9145, 8942, 94828, 77821, 67340, 60689, 56380, 53541, 51644, 50361, 49484, 562595

Offset

2,2

Comments

Number of rises s_{k+1} > s_k in an RGS [s_1, ..., s_n] of a set partition of {1, ..., n}, where s_i is the subset containing i, 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 [apparently unpublished, Joerg Arndt, Mar 05 2016]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

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 3 rises, at i = 1, i = 3 and i = 5.
1;
3,2;
10,7,6;
37,27,23,21;
151,114,97,88,83;
674,523,446,403,378,363;
3263,2589,2217,1999,1867,1785,1733;
17007,13744,11829,10658,9923,9452,9145,8942;
94828,77821,67340,60689,56380,53541,51644,50361,49484;
562595,467767,406953,367101,340551,322619,310365,301905,296011,291871;
3535027,2972432,2599493,2348182,2176575,2058068,1975425,1917290,1876075, 1846648,1825501;

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_] := BellB[n] - BellB[n-1] - 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

Cf. Bell numbers A005493, A011965.

Cf. A056857-A056863.

Sequence in context: A227631 A246830 A268531 * A302846 A214844 A214966

Adjacent sequences: A056858 A056859 A056860 * A056862 A056863 A056864

Keyword

easy,nonn,tabl

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

Several terms corrected, R. J. Mathar, Mar 08 2016

Status

approved