A276922 Number T(n,k) of ordered set partitions of [n] where the maximal block size equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 24, 42, 8, 1, 0, 120, 330, 80, 10, 1, 0, 720, 2970, 860, 120, 12, 1, 0, 5040, 30240, 10290, 1540, 168, 14, 1, 0, 40320, 345240, 136080, 21490, 2464, 224, 16, 1, 0, 362880, 4377240, 1977360, 326970, 38808, 3696, 288, 18, 1

Offset: 0

Author

Alois P. Heinz, Sep 22 2016

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
  1;
  0,     1;
  0,     2,      1;
  0,     6,      6,      1;
  0,    24,     42,      8,     1;
  0,   120,    330,     80,    10,    1;
  0,   720,   2970,    860,   120,   12,   1;
  0,  5040,  30240,  10290,  1540,  168,  14,  1;
  0, 40320, 345240, 136080, 21490, 2464, 224, 16, 1;
  ...
		

Links

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

Crossrefs

Columns k=0-10 give: A000007, A000142 (for n>0), A320758, A320759, A320760, A320761, A320762, A320763, A320764, A320765, A320766.

Row sums give A000670.

T(2n,n) gives A276923.

Cf. A080510, A276921.

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, add(
       A(n-i, k)*binomial(n, i), i=1..min(n, k)))
    end:
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
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 - i, k]*Binomial[n, i], {i, 1, Min[n, k]}]]; T[n_, k_] :=  A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[T[n, k], {n, 0, 10}, { k, 0, n}] // Flatten (* Jean-François Alcover, Feb 11 2017, translated from Maple *)

Formula

E.g.f. for column k>0: 1/(1-Sum_{i=1..k} x^i/i!) - 1/(1-Sum_{i=1..k-1} x^i/i!).

T(n,k) = A276921(n,k) - A276921(n,k-1) for k>0. T(n,0) = A000007(0).