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

0,5

Links

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

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).

Example

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;
  ...

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 *)

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.

Sequence in context: A111184 A111596 A271703 * A129062 A281662 A163936

Adjacent sequences: A276919 A276920 A276921 * A276923 A276924 A276925

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 22 2016

Status

approved