A271206 -id:A271206 - cp's OEIS frontend

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

Brian Drake, Feb 08 2011

			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;
  ...

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.

Columns k=0-10 give: A000012, A271788, A271789, A271790, A271791, A271792, A271793, A271794, A271795, A271796, A271797.
Row sums give A000110.
T(n+1,n-1) gives A000217.
T(2n,n) gives A271841.
Cf. A185983, A270953, A271206, A271270, A271271, A272064.

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

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 *)

More terms from Alois P. Heinz, Oct 11 2011

A271207 Number of set partitions of [n] avoiding triples (t,t+1,t+2) having t+i in block b+i for some b.

Original entry on oeis.org

1, 1, 2, 4, 10, 28, 89, 315, 1233, 5285, 24583, 123062, 658335, 3741625, 22483415, 142264589, 944652336, 6562959239, 47583055191, 359196368057, 2817394708454, 22919157785777, 193045807856919, 1681025045594032, 15112573721911697, 140088106892677440

Offset: 0

Alois P. Heinz, Apr 01 2016

nonn

			a(4) = 10: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4.

Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Partition of a set

Column k=0 of A271206.

b:= proc(n, i, t, m) option remember; `if`(n=0, 1, add(`if`(
     t and j=i+1, 0, b(n-1, j, is(j=i+1), max(m, j))), j=1..m+1))
    end:
a:= n-> b(n, 1, false, 0):
seq(a(n), n=0..30);

b[n_, i_, t_, m_] := b[n, i, t, m] = If[n == 0, 1, Sum[If[t && j == i + 1, 0, b[n - 1, j, j == i + 1, Max[m, j]]], {j, 1, m + 1}]];
a[n_] := b[n, 1, False, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 27 2018, translated from Maple *)

cp's OEIS Frontend

A185982 Triangle read by rows: number of set partitions of n elements with k connectors, 0<=k

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