login
Triangle T(n,k) is the number of restricted growth strings (RGS) of set partitions of {1..n} that have a decrease at index k (1<=k<n).
2

%I #28 May 23 2016 02:46:24

%S 0,0,1,0,3,4,0,10,14,16,0,37,54,63,68,0,151,228,271,296,311,0,674,

%T 1046,1264,1396,1478,1530,0,3263,5178,6349,7084,7555,7862,8065,0,

%U 17007,27488,34139,38448,41287,43184,44467,45344,0,94828,155642,195494,222044,239976,252230,260690,266584,270724

%N Triangle T(n,k) is the number of restricted growth strings (RGS) of set partitions of {1..n} that have a decrease at index k (1<=k<n).

%C 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).

%C 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

%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 [apparently unpublished, _Joerg Arndt_, Mar 05 2016]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="/A056862/b056862.txt">Rows n = 2..100, flattened</a>

%F T(n,k) = B(n) - B(n-1) - A056861(n,k). - _Franklin T. Adams-Watters_, Jun 08 2006

%F Conjecture: T(n,3) = 2*A011965(n). - _R. J. Mathar_, Mar 08 2016

%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 1 fall, at i = 2.

%e 0;

%e 0,1;

%e 0,3,4;

%e 0,10,14,16;

%e 0,37,54,63,68;

%e 0,151,228,271,296,311;

%e 0,674,1046,1264,1396,1478,1530;

%e 0,3263,5178,6349,7084,7555,7862,8065;

%e 0,17007,27488,34139,38448,41287,43184,44467,45344;

%e 0,94828,155642,195494,222044,239976,252230,260690,266584,270724;

%e 0,562595,935534,1186845,1358452,1476959,1559602,1617737,1658952,1688379, 1709526;

%p b:= proc(n, i, m, t) option remember; `if`(n=0, [1, 0],

%p add((p-> p+[0, `if`(j<i, p[1]*x^t, 0)])(

%p b(n-1, j, max(m, j), t+1)), j=1..m+1))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=1..n-1))(b(n, 1, 0$2)[2]):

%p seq(T(n), n=2..12); # _Alois P. Heinz_, Mar 24 2016

%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[n-1, j, Max[m, j], t+1]], {j, 1, m+1}]];

%t T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n - 1}]][b[n, 1, 0, 0][[2]]];

%t Table[T[n], {n, 2, 12}] // Flatten (* _Jean-François Alcover_, May 23 2016, after _Alois P. Heinz_ *)

%Y Cf. Bell numbers A005493.

%Y Cf. A056857-A056863.

%K easy,nonn,tabl

%O 2,5

%A Winston C. Yang (winston(AT)cs.wisc.edu), Aug 31 2000

%E Edited and extended by _Franklin T. Adams-Watters_, Jun 08 2006

%E Clarified definition and edited comment and example, _Joerg Arndt_, Mar 08 2016

%E Data corrected, _R. J. Mathar_, Mar 08 2016