login
A258280
Number T(n,k) of partitions of k copies of n into distinct parts; triangle T(n,k), n>=0, 0<=k<=max(1,ceiling(n/2)), read by rows.
12
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 1, 1, 5, 7, 4, 1, 1, 6, 9, 5, 1, 1, 8, 16, 13, 5, 1, 1, 10, 21, 18, 7, 1, 1, 12, 33, 37, 20, 6, 1, 1, 15, 46, 56, 31, 8, 1, 1, 18, 68, 103, 75, 29, 7, 1, 1, 22, 95, 154, 118, 47, 10, 1, 1, 27, 140, 279, 266, 134, 40, 8, 1
OFFSET
0,8
LINKS
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
Row sums give 1 + A258289.
Row lengths give 1 + A065033.
T(n^2,n) gives A284824.
Sequence in context: A343662 A152198 A259342 * A159255 A376647 A035693
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, May 25 2015
STATUS
approved