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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 (list; graph; refs; listen; history; text; internal format)
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 A035693 A328818
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, May 25 2015
STATUS
approved

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Last modified July 19 10:32 EDT 2024. Contains 374392 sequences. (Running on oeis4.)