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

0,5

Links

Alois P. Heinz, Rows n = 0..140, flattened

Wikipedia, Partition of a set

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!.

Example

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;
  ...

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 *)

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.

Sequence in context: A274086 A255982 A256061 * A002652 A202541 A070676

Adjacent sequences: A323096 A323097 A323098 * A323100 A323101 A323102

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 30 2019

Status

approved