A262071 Number T(n,k) of ordered partitions of an n-set with nondecreasing block sizes and maximal block size equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 1, 0, 6, 3, 1, 0, 24, 18, 4, 1, 0, 120, 90, 30, 5, 1, 0, 720, 630, 200, 45, 6, 1, 0, 5040, 4410, 1610, 350, 63, 7, 1, 0, 40320, 37800, 13440, 3290, 560, 84, 8, 1, 0, 362880, 340200, 130200, 30870, 5922, 840, 108, 9, 1, 0, 3628800, 3515400, 1327200, 334950, 61992, 9870, 1200, 135, 10, 1

Offset: 0

Author

Alois P. Heinz, Sep 10 2015

Keywords

• nonn
• tabl

Examples

			T(3,1) = 6: 1|2|3, 1|3|2, 2|1|3, 2|3|1, 3|1|2, 3|2|1.
T(3,2) = 3: 1|23, 2|13, 3|12.
T(3,3) = 1: 123.
Triangle T(n,k) begins:
  1;
  0,     1;
  0,     2,     1;
  0,     6,     3,     1;
  0,    24,    18,     4,    1;
  0,   120,    90,    30,    5,   1;
  0,   720,   630,   200,   45,   6,  1;
  0,  5040,  4410,  1610,  350,  63,  7, 1;
  0, 40320, 37800, 13440, 3290, 560, 84, 8, 1;
		

Links

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

Crossrefs

Columns k=0-10 give: A000007, A000142 (for n>0), A272492, A272493, A272494, A272495, A272496, A272497, A272498, A272499, A272500.

Main diagonal gives A000012.

Row sums give A005651.

T(2n,n) gives A266518.

Cf. A262072.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+`if`(i>n, 0, binomial(n, i)*b(n-i, i))))
    end:
T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, Binomial[n, i]*b[n - i, i]]]]; T[n_, k_] :=  b[n, k] - If[k == 0, 0, b[n, k - 1]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 12 2016, Alois P. Heinz *)

Formula

E.g.f. of column k: x^k * Product_{i=1..k} (i-1)!/(i!-x^i).