A086659 T(n,k) counts the set partitions of n containing k-1 blocks of length 1.

1, 1, 3, 4, 4, 6, 11, 20, 10, 10, 41, 66, 60, 20, 15, 162, 287, 231, 140, 35, 21, 715, 1296, 1148, 616, 280, 56, 28, 3425, 6435, 5832, 3444, 1386, 504, 84, 36, 17722, 34250, 32175, 19440, 8610, 2772, 840, 120, 45, 98253, 194942, 188375, 117975, 53460, 18942, 5082, 1320, 165, 55

Offset: 2

Author

Wouter Meeussen, Jul 27 2003

Keywords

• nonn
• tabl
• easy

Examples

			The 15 set partitions of {1,2,3,4} consist of 4 partitions with 0 blocks of length 1 : {{1,2,3,4}},{{1,2},{3,4}},{{1,3},{2,4}},{{1,4},{2,3}},
4 partitions with 1 block of length 1 : {{1},{2,3,4}},{{1,2,3},{4}},{{1,2,4},{3}},{{1,3,4},{2}}
6 partitions with 2 blocks of length 1 : {{1},{2},{3,4}},{{1},{2,3},{4}},{{1},{2,4},{3}},{{1,2},{3},{4}},{{1,3},{2},{4}},{{1,4},{2},{3}}.
(There are no partitions with n-1 blocks of length 1 and 1 with n of them)
    1;
    1,   3;
    4,   4,   6;
   11,  20,  10,  10;
   41,  66,  60,  20, 15;
  162, 287, 231, 140, 35, 21;
  ...
		

Links

• Alois P. Heinz, Rows n = 2..142, flattened

Crossrefs

Row sums = Bell[n]-1 (A058692), first column=A000296, main diagonal = triangular numbers A000217.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(i=1, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-2))(b(n$2)):
seq(T(n), n=2..16);  # Alois P. Heinz, Mar 08 2015

Mathematica

Table[Count[Count[ #, {_Integer}]&/@SetPartitions[n], # ]&/@Range[0, n-2], {n, 2, 10}]

Formula

E.g.f.: exp(x*y)*(exp(exp(x)-1-x)-1). - Vladeta Jovovic, Jul 28 2003

Extensions

More terms from Vladeta Jovovic, Jul 28 2003