A271023 Number T(n,k) of set partitions of [n] having exactly k pairs (i,j) with i=0, 0<=k<=n*(n-1)/2 read by rows.

1, 1, 1, 1, 1, 3, 0, 1, 1, 6, 3, 4, 0, 0, 1, 1, 10, 15, 10, 10, 0, 5, 0, 0, 0, 1, 1, 15, 45, 35, 60, 0, 25, 15, 0, 0, 6, 0, 0, 0, 0, 1, 1, 21, 105, 140, 210, 105, 105, 105, 0, 35, 21, 21, 0, 0, 0, 7, 0, 0, 0, 0, 0, 1, 1, 28, 210, 476, 665, 840, 350, 700, 210

Offset: 0

Author

Alois P. Heinz, Mar 28 2016

Keywords

• nonn
• look
• tabf

Examples

			T(3,0) = 1: 1|2|3.
T(3,1) = 3: 12|3, 13|2, 1|23.
T(3,3) = 1: 123.
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  3,  0,  1;
  1,  6,  3,  4,  0, 0,  1;
  1, 10, 15, 10, 10, 0,  5,  0, 0, 0, 1;
  1, 15, 45, 35, 60, 0, 25, 15, 0, 0, 6, 0, 0, 0, 0, 1;
		

Links

• Alois P. Heinz, Rows n = 0..40, flattened

Crossrefs

Columns k=0-2 give: A000012, A161680, A050534(n-1) for n>0.

Row sums give A000110.

Cf. A105488, A161680, A271024.

Programs

Maple

b:= proc(n, l) option remember; `if`(n=0, x^
      add(j*(j-1)/2, j=l), b(n-1, [l[], 1])+
      add(b(n-1, subsop(j=l[j]+1, l)), j=1..nops(l)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [])):
seq(T(n), n=0..10);

Mathematica

b[n_, l_] := b[n, l] = If[n == 0, x^Sum[j*(j-1)/2, {j, l}], b[n-1, Append[l, 1]] + Sum[b[n-1, ReplacePart[l, j -> l[[j]]+1]], {j, 1, Length[l]}]];
T[n_] := CoefficientList[b[n, {}], x];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 29 2022, after Alois P. Heinz *)

Formula

T(n,k) = A271024(n,n*(n-1)/2-k).

Sum_{k=0..n*(n-1)/2} k * T(n,k) = A105488(n+2) for n > 1.