A372762 Number T(n,k) of partitions of [n] having exactly k blocks of minimal size; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 5, 9, 0, 1, 0, 31, 10, 10, 0, 1, 0, 82, 70, 35, 15, 0, 1, 0, 344, 336, 140, 35, 21, 0, 1, 0, 1661, 1393, 616, 385, 56, 28, 0, 1, 0, 7942, 6210, 4984, 1386, 504, 84, 36, 0, 1, 0, 38721, 41331, 22590, 8610, 3717, 840, 120, 45, 0, 1

Offset

0,8

Links

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

Wikipedia, Partition of a set

Formula

Sum_{k=0..n} k * T(n,k) = A372650(n).

Example

T(5,1) = 31: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34.
T(5,2) = 10: 123|4|5, 124|3|5, 125|3|4, 134|2|5, 135|2|4, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345.
T(5,3) = 10: 12|3|4|5, 13|2|4|5, 1|23|4|5, 14|2|3|5, 1|24|3|5, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
T(5,4) = 0.
T(5,5) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
  1;
  0,     1;
  0,     1,     1;
  0,     4,     0,     1;
  0,     5,     9,     0,    1;
  0,    31,    10,    10,    0,    1;
  0,    82,    70,    35,   15,    0,   1;
  0,   344,   336,   140,   35,   21,   0,   1;
  0,  1661,  1393,   616,  385,   56,  28,   0,  1;
  0,  7942,  6210,  4984, 1386,  504,  84,  36,  0, 1;
  0, 38721, 41331, 22590, 8610, 3717, 840, 120, 45, 0, 1;
  ...

Maple

b:= proc(n, m, t) option remember; `if`(n=0, x^t,
      add(binomial(n-1, j-1)*b(n-j, min(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$2, 0)):
seq(T(n), n=0..12);

Crossrefs

Columns k=0-2 give: A000007, A224219, A372764.

Row sums give A000110.

T(2n,n) gives A271425.

Cf. A372650, A372722.

Sequence in context: A147312 A352771 A271423 * A019974 A344373 A325873

Adjacent sequences: A372759 A372760 A372761 * A372763 A372764 A372765

Keyword

nonn,tabl

Author

Alois P. Heinz, May 12 2024

Status

approved