A323099 Number T(n,k) of colored set partitions of [n] where exactly k colors are used for the elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 4, 0, 5, 30, 30, 0, 15, 210, 540, 360, 0, 52, 1560, 7800, 12480, 6240, 0, 203, 12586, 109620, 316680, 365400, 146160, 0, 877, 110502, 1583862, 7366800, 14733600, 13260240, 4420080, 0, 4140, 1051560, 23995440, 169011360, 521640000, 792892800, 584236800, 166924800

Offset: 0

Author

Alois P. Heinz, Aug 30 2019

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,      4;
  0,   5,     30,      30;
  0,  15,    210,     540,     360;
  0,  52,   1560,    7800,   12480,     6240;
  0, 203,  12586,  109620,  316680,   365400,   146160;
  0, 877, 110502, 1583862, 7366800, 14733600, 13260240, 4420080;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Wikipedia, Partition of a set

Crossrefs

Columns k=0-1 give: A000007, A000110 (for n>0).

Row sums give A121017.

Main diagonal gives A137341.

Cf. A000142, A008277, A048993, A019538, A131689.

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(n-j, k)*binomial(n-1, j-1)*k^j, j=1..n))
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);
# second Maple program:
T:= (n, k)-> combinat[bell](n)*Stirling2(n,k)*k!:
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-j, k] Binomial[n-1, j-1] k^j, {j, 1, n}]];
T[n_, k_] := Sum[A[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten
(* second program: *)
T[n_, k_] := BellB[n] StirlingS2[n, k] k!;
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

Formula

T(n,k) = Bell(n) * Sum_{i=0..k} (k-i)^n * (-1)^i * C(k,i).

T(n,k) = Bell(n) * A131689(n,k).

T(n,k) = Bell(n) * Stirling2(n,k) * k!.