%I
%S 1,3,2,10,7,6,37,27,23,21,151,114,97,88,83,674,523,446,403,378,363,
%T 3263,2589,2217,1999,1867,1785,1733,17007,13744,11829,10658,9923,9452,
%U 9145,8942,94828,77821,67340,60689,56380,53541,51644,50361,49484,562595
%N 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).
%C 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).
%C Note that the number of equalities at any index is B(n1), where B(n) are the Bell numbers.  _Franklin T. AdamsWatters_, Jun 08 2006
%D W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000. [apparently unpublished, _Joerg Arndt_, Mar 05 2016]
%H Alois P. Heinz, <a href="/A056861/b056861.txt">Rows n = 2..100, flattened</a>
%e 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.
%e 1;
%e 3,2;
%e 10,7,6;
%e 37,27,23,21;
%e 151,114,97,88,83;
%e 674,523,446,403,378,363;
%e 3263,2589,2217,1999,1867,1785,1733;
%e 17007,13744,11829,10658,9923,9452,9145,8942;
%e 94828,77821,67340,60689,56380,53541,51644,50361,49484;
%e 562595,467767,406953,367101,340551,322619,310365,301905,296011,291871;
%e 3535027,2972432,2599493,2348182,2176575,2058068,1975425,1917290,1876075, 1846648,1825501;
%t 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[n1, j, Max[m, j], t+1]], {j, 1, m+1}]];
%t T[n_] := BellB[n]  BellB[n1]  Function[p, Table[Coefficient[p, x, i], {i, 1, n1}]][b[n, 1, 0, 0][[2]]];
%t Table[T[n], {n, 2, 12}] // Flatten (* _JeanFrançois Alcover_, May 23 2016, after _Alois P. Heinz_ *)
%Y Cf. Bell numbers A005493, A011965.
%Y Cf. A056857A056863.
%K easy,nonn,tabl
%O 2,2
%A Winston C. Yang (winston(AT)cs.wisc.edu), Aug 31 2000
%E Edited and extended by _Franklin T. AdamsWatters_, Jun 08 2006
%E Clarified definition and edited comment and example, _Joerg Arndt_, Mar 08 2016
%E Several terms corrected, _R. J. Mathar_, Mar 08 2016
