A086659 - OEIS

%I #16 Jan 05 2022 18:16:14

%S 1,1,3,4,4,6,11,20,10,10,41,66,60,20,15,162,287,231,140,35,21,715,

%T 1296,1148,616,280,56,28,3425,6435,5832,3444,1386,504,84,36,17722,

%U 34250,32175,19440,8610,2772,840,120,45,98253,194942,188375,117975,53460,18942,5082,1320,165,55

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

%H Alois P. Heinz, <a href="/A086659/b086659.txt">Rows n = 2..142, flattened</a>

%F E.g.f.: exp(x*y)*(exp(exp(x)-1-x)-1). - _Vladeta Jovovic_, Jul 28 2003

%e 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}},

%e 4 partitions with 1 block of length 1 : {{1},{2,3,4}},{{1,2,3},{4}},{{1,2,4},{3}},{{1,3,4},{2}}

%e 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}}.

%e (There are no partitions with n-1 blocks of length 1 and 1 with n of them)

%e     1;

%e     1,   3;

%e     4,   4,   6;

%e    11,  20,  10,  10;

%e    41,  66,  60,  20, 15;

%e   162, 287, 231, 140, 35, 21;

%e   ...

%p with(combinat):

%p b:= proc(n, i) option remember; expand(`if`(n=0, 1,

%p       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*

%p       b(n-i*j, i-1)*`if`(i=1, x^j, 1), j=0..n/i))))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n-2))(b(n$2)):

%p seq(T(n), n=2..16);  # _Alois P. Heinz_, Mar 08 2015

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

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

%K nonn,tabl,easy

%O 2,3

%A _Wouter Meeussen_, Jul 27 2003

%E More terms from _Vladeta Jovovic_, Jul 28 2003