OFFSET
0,8
LINKS
Alois P. Heinz, Rows n = 0..37, flattened
FORMULA
T(n,k) = 1/k! * [Product_{i=1..k} x_i^n] Product_{j>0} (1+Sum_{i=1..k} x_i^j).
EXAMPLE
T(7,0) = 1: [].
T(7,1) = 5: [7], [6,1], [5,2], [4,3], [4,2,1].
T(7,2) = 7: [7;6,1], [7;5,2], [7;4,3], [7;4,2,1], [6,1;5,2], [6,1;4,3], [5,2;4,3].
T(7,3) = 4: [7;6,1;5,2], [7;6,1;4,3], [7;5,2;4,3], [6,1;5,2;4,3].
T(7,4) = 1: [7;6,1;5,2;4,3].
T(8,4) = 1: [8;7,1;6,2;5,3].
Triangle T(n,k) begins:
00 : 1, 1;
01 : 1, 1;
02 : 1, 1;
03 : 1, 2, 1;
04 : 1, 2, 1;
05 : 1, 3, 3, 1;
06 : 1, 4, 4, 1;
07 : 1, 5, 7, 4, 1;
08 : 1, 6, 9, 5, 1;
09 : 1, 8, 16, 13, 5, 1;
10 : 1, 10, 21, 18, 7, 1;
11 : 1, 12, 33, 37, 20, 6, 1;
12 : 1, 15, 46, 56, 31, 8, 1;
13 : 1, 18, 68, 103, 75, 29, 7, 1;
14 : 1, 22, 95, 154, 118, 47, 10, 1;
15 : 1, 27, 140, 279, 266, 134, 40, 8, 1;
...
MAPLE
b:= proc() option remember; local m; m:= args[nargs];
`if`(nargs=1, 1, `if`(args[1]=0, b(args[t] $t=2..nargs),
`if`(m=0 or add(args[i], i=1..nargs-1)> m*(m+1)/2, 0,
b(args[t] $t=1..nargs-1, m-1) +add(`if`(args[j]-m<0, 0,
b(sort([seq(args[i] -`if`(i=j, m, 0), i=1..nargs-1)])[]
, m-1)), j=1..nargs-1))))
end:
T:= (n, k)-> b(n$k+1)/k!:
seq(seq(T(n, k), k=0..max(1, ceil(n/2))), n=0..15);
MATHEMATICA
disParts[n_] := disParts[n] = Select[IntegerPartitions[n], Length[#] == Length[Union[#]]&];
T[n_, k_] := Select[Subsets[disParts[n], {k}], Length[Flatten[#]] == Length[Union[Flatten[#]]]&] // Length;
Table[T[n, k], {n, 0, 15}, {k, 0, Max[1, Ceiling[n/2]]}] // Flatten (* Jean-François Alcover, Feb 17 2021 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, May 25 2015
STATUS
approved