A322670 Number T(n,k) of colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 3, 0, 1, 12, 16, 0, 1, 41, 156, 131, 0, 1, 140, 1155, 2460, 1496, 0, 1, 497, 8020, 32600, 47355, 22482, 0, 1, 1848, 55629, 385420, 1004360, 1098678, 426833, 0, 1, 7191, 394884, 4396189, 18304510, 34625304, 30259712, 9934563

Offset: 0

Author

Alois P. Heinz, Aug 29 2019

Keywords

• nonn
• tabl

Examples

			T(3,2) = 12: 1a|2a3b, 1b|2a3b, 1a3b|2a, 1a3b|2b, 1a2b|3a, 1a2b|3b, 1a|2a|3b, 1a|2b|3a, 1b|2a|3a, 1a|2b|3b, 1b|2a|3b, 1b|2b|3a.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,    3;
  0, 1,   12,    16;
  0, 1,   41,   156,    131;
  0, 1,  140,  1155,   2460,    1496;
  0, 1,  497,  8020,  32600,   47355,   22482;
  0, 1, 1848, 55629, 385420, 1004360, 1098678, 426833;
  ...
		

Links

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

Crossrefs

Columns k=0-2 give: A000007, A057427, A325482.

Main diagonal gives A023998.

Row sums give A325478.

T(2n,n) gives A325481.

Cf. A321296, A323128, A325930.

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*
      binomial(n-1, j-1)*binomial(k, j), j=1..min(k, 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);

Mathematica

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[A[n-j, k] Binomial[n-1, j-1]* Binomial[k, j], {j, 1, Min[k, n]}]];
T[n_, k_] := Sum[A[n, k-i] (-1)^i Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

Formula

Sum_{k=1..n} k * T(n,k) = A325930(n).