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

0,5

Links

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

Wikipedia, Partition of a set

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

Example

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;
  ...

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 *)

Prog

(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

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.

Sequence in context: A083417 A021479 A073583 * A060136 A327358 A256664

Adjacent sequences: A324159 A324160 A324161 * A324163 A324164 A324165

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 02 2019

Status

approved