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A185982 Triangle read by rows: number of set partitions of n elements with k connectors, 0<=k<n. 21
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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.

EXAMPLE

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;

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-Fran├žois Alcover, Apr 13 2016, after Alois P. Heinz *)

CROSSREFS

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.

Sequence in context: A008277 A218577 A193387 * A263858 A263862 A263861

Adjacent sequences:  A185979 A185980 A185981 * A185983 A185984 A185985

KEYWORD

nonn,tabl

AUTHOR

Brian Drake, Feb 08 2011

EXTENSIONS

More terms from Alois P. Heinz, Oct 11 2011

STATUS

approved

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Last modified November 15 19:54 EST 2018. Contains 317240 sequences. (Running on oeis4.)