A324162 Number T(n,k) of set partitions of [n] where each subset is again partitioned into k nonempty subsets; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 1, 0, 5, 3, 1, 0, 15, 10, 6, 1, 0, 52, 45, 25, 10, 1, 0, 203, 241, 100, 65, 15, 1, 0, 877, 1428, 511, 350, 140, 21, 1, 0, 4140, 9325, 3626, 1736, 1050, 266, 28, 1, 0, 21147, 67035, 29765, 9030, 6951, 2646, 462, 36, 1, 0, 115975, 524926, 250200, 60355, 42651, 22827, 5880, 750, 45, 1

Offset: 0

Author

Alois P. Heinz, Sep 02 2019

Keywords

• nonn
• tabl

Examples

			T(4,2) = 10: 123/4, 124/3, 12/34, 134/2, 13/24, 14/23, 1/234, 1/2|3/4, 1/3|2/4, 1/4|2/3.
Triangle T(n,k) begins:
  1;
  0,    1;
  0,    2,    1;
  0,    5,    3,    1;
  0,   15,   10,    6,    1;
  0,   52,   45,   25,   10,    1;
  0,  203,  241,  100,   65,   15,   1;
  0,  877, 1428,  511,  350,  140,  21,  1;
  0, 4140, 9325, 3626, 1736, 1050, 266, 28, 1;
  ...
		

Links

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

Crossrefs

Columns k=0-10 give: A000007, A000110 (for n>0), A060311, A327504, A327505, A327506, A327507, A327508, A327509, A327510, A327511.

Row sums give A324238.

T(2n,n) gives A324241.

Cf. A008277, A048993, A039810, A130191, A325929.

Programs

Maple

T:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0, add(
      T(n-j, k)*binomial(n-1, j-1)*Stirling2(j, k), j=k..n)))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

nmax = 10;
col[k_] := col[k] = CoefficientList[Exp[(Exp[x]-1)^k/k!] + O[x]^(nmax+1), x][[k+1;;]] Range[k, nmax]!;
T[n_, k_] := Which[k == n, 1, k == 0, 0, True, col[k][[n-k+1]]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 26 2020 *)

PARI

T(n, k) = if(k==0, 0^n, sum(j=0, n\k, (k*j)!*stirling(n, k*j, 2)/(k!^j*j!))); \\ Seiichi Manyama, May 07 2022

Formula

E.g.f. of column k>0: exp((exp(x)-1)^k/k!).

Sum_{k=1..n} k * T(n,k) = A325929(n).

T(n,k) = Sum_{j=0..floor(n/k)} (k*j)! * Stirling2(n,k*j)/(k!^j * j!) for k > 0. - Seiichi Manyama, May 07 2022