A185982
Triangle read by rows: number of set partitions of n elements with k connectors, 0<=k
1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 16, 24, 10, 1, 1, 39, 86, 61, 15, 1, 1, 105, 307, 313, 129, 21, 1, 1, 314, 1143, 1520, 891, 242, 28, 1, 1, 1035, 4513, 7373, 5611, 2161, 416, 36, 1, 1, 3723, 18956, 36627, 34213, 17081, 4658, 670, 45, 1, 1, 14494, 84546, 188396, 208230, 127540, 45095, 9187, 1025, 55, 1
Offset: 1
Examples
A connector is a pair (a, a+1) in a set partition if a is in block i and a+1 is in block i+1, for some i. For example a(4,1) = 7, counting 1/234, 13/2/4, 14/23, 134/2, 12/34, 124/3, 123/4. Triangle begins: 1; 1, 1; 1, 3, 1; 1, 7, 6, 1; 1, 16, 24, 10, 1; 1, 39, 86, 61, 15, 1; 1, 105, 307, 313, 129, 21, 1; ...
Links
- Alois P. Heinz, Rows n = 1..141, flattened
- T. Mansour and A. O. Munagi, Block-connected set partitions, European J. Combin., 31 (2010), 887-902.
Crossrefs
Programs
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Maple
b:= proc(n, i, m) option remember; `if`(n=0, 1, add(expand( b(n-1, j, max(m, j))*`if`(j=i+1, x, 1)), j=1..m+1)) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n, 1, 0)): seq(T(n), n=1..12); # Alois P. Heinz, Mar 25 2016 -
Mathematica
b[n_, i_, m_] := b[n, i, m] = If[n == 0, 1, Sum[b[n-1, j, Max[m, j]]*If[j == i+1, x, 1], {j, 1, m+1}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[n, 1, 0]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Apr 13 2016, after Alois P. Heinz *)
Extensions
More terms from Alois P. Heinz, Oct 11 2011