login
Triangle of number of rises in restricted growth strings (RGS) for the set partitions of n.
3

%I #22 May 23 2016 02:46:08

%S 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,

%T 1,1,28,322,1268,1709,732,79,1,1,36,588,3571,8136,6794,1887,133,1,1,

%U 45,1002,8878,31532,44924,24717,4654,221,1,1,55,1617,20053,104927,234412,221857,84170,11113,364,1

%N Triangle of number of rises in restricted growth strings (RGS) for the set partitions of n.

%C 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.

%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 [apparently unpublished, _Joerg Arndt_, Mar 05 2016]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="/A056858/b056858.txt">Rows n = 1..100, flattened</a>

%e 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.

%e 1;

%e 1,1;

%e 1,3,1;

%e 1,6,7,1;

%e 1,10,26,14,1;

%e 1,15,71,89,26,1;

%e 1,21,161,380,267,46,1;

%e 1,28,322,1268,1709,732,79,1;

%e 1,36,588,3571,8136,6794,1887,133,1;

%e 1,45,1002,8878,31532,44924,24717,4654,221,1;

%e 1,55,1617,20053,104927,234412,221857,84170,11113,364,1;

%e 1,66,2497,41965,310255,1025377,1528351,1006028,272557,25903,596,1;

%p b:= proc(n, i, m) option remember; expand(

%p `if`(n=0, x, add(b(n-1, j, max(m, j))*

%p `if`(j>i, x, 1), j=1..m+1)))

%p end:

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

%p seq(T(n), n=1..12); # _Alois P. Heinz_, Mar 24 2016

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

%t Table[T[n], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, May 23 2016, after _Alois P. Heinz_ *)

%Y Cf. A000110 (row sums).

%Y Cf. A056857-A056863.

%Y Column 1 is triangular numbers (A000217); diagonal T(n, n-1) appears to be A001924.

%K easy,nonn,tabl

%O 1,5

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

%E More terms from _Franklin T. Adams-Watters_, Jun 08 2006

%E Clarified definition and edited comment and example, _Joerg Arndt_, Mar 05 2016