OFFSET
0,6
LINKS
Alois P. Heinz, Rows n = 0..33, flattened
Wikipedia, Partition of a set
EXAMPLE
T(4,0) = 1: 1|2|3|4.
T(4,1) = 3: 12|3|4, 1|23|4, 1|2|34.
T(4,2) = 5: 123|4, 12|34, 13|2|4, 1|234, 1|24|3.
T(4,3) = 4: 1234, 124|3, 134|2, 14|2|3.
T(4,4) = 2: 13|24, 14|23.
T(5,5) = 8: 124|35, 125|34, 13|245, 13|25|4, 145|23, 15|23|4, 14|2|35, 15|2|34.
T(5,6) = 6: 134|25, 135|24, 14|235, 15|234, 14|25|3, 15|24|3.
T(6,9) = 6: 14|25|36, 14|26|35, 15|24|36, 16|24|35, 15|26|34, 16|25|34.
Triangle T(n,k) begins:
1;
1;
1, 1;
1, 2, 2;
1, 3, 5, 4, 2;
1, 4, 9, 12, 12, 8, 6;
1, 5, 14, 25, 34, 36, 36, 28, 18, 6;
1, 6, 20, 44, 74, 100, 122, 132, 132, 108, 78, 36, 24;
...
MAPLE
b:= proc(n, m) option remember; `if`(n=0, x^add(-i, i=m), add(
b(n-1, subs(j=n, m)), j=m)+expand(b(n-1, {m[], n})*x^n))
end:
T:= (n, k)-> coeff(b(n, {}), x, k):
seq(seq(T(n, k), k=0..(h-> h*(n-h))(iquo(n, 2))), n=0..10);
# second Maple program:
b:= proc(n, s) option remember; `if`(n=0, 1, (k-> `if`(n>k,
b(n-1, s)*(k+1), 0)+`if`(n>k+1, b(n-1, {s[], n}), 0)+
add(expand(x^(h-n)*b(n-1, s minus {h})), h=s))(nops(s)))
end:
T:= (n, k)-> coeff(b(n, {}), x, k):
seq(seq(T(n, k), k=0..floor(n^2/4)), n=0..10);
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Dec 21 2023
STATUS
approved