A372722 Number T(n,k) of partitions of [n] having exactly k blocks of maximal size; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 11, 3, 0, 1, 0, 36, 15, 0, 0, 1, 0, 132, 55, 15, 0, 0, 1, 0, 596, 175, 105, 0, 0, 0, 1, 0, 2809, 805, 420, 105, 0, 0, 0, 1, 0, 14608, 4053, 1540, 945, 0, 0, 0, 0, 1, 0, 79448, 24906, 5950, 4725, 945, 0, 0, 0, 0, 1, 0, 461748, 151371, 37730, 17325, 10395, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Alois P. Heinz, May 11 2024

Keywords

• nonn
• tabl

Examples

			T(4,1) = 11: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
T(4,2) = 3: 12|34, 13|24, 14|23.
T(4,3) = 0.
T(4,4) = 1: 1|2|3|4.
Triangle T(n,k) begins:
  1;
  0,     1;
  0,     1,     1;
  0,     4,     0,    1;
  0,    11,     3,    0,    1;
  0,    36,    15,    0,    0,   1;
  0,   132,    55,   15,    0,   0, 1;
  0,   596,   175,  105,    0,   0, 0, 1;
  0,  2809,   805,  420,  105,   0, 0, 0, 1;
  0, 14608,  4053, 1540,  945,   0, 0, 0, 0, 1;
  0, 79448, 24906, 5950, 4725, 945, 0, 0, 0, 0, 1;
  ...
		

Links

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

Crossrefs

Columns k=0-1 give: A000007, A372721.

Row sums give A000110.

T(2n,n) gives A001147.

T(3n,n) gives A271715.

Cf. A372649, A372762.

Programs

Maple

b:= proc(n, m, t) option remember; `if`(n=0, x^t,
      add(binomial(n-1, j-1)*b(n-j, max(j, m),
     `if`(j>m, 1, `if`(j=m, t+1, t))), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0$2)):
seq(T(n), n=0..12);

Formula

Sum_{k=0..n} k * T(n,k) = A372649(n).