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.

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

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

Columns k=0-10 give: A000012, A000009, A108796, A258281, A258282, A258283, A258284, A258285, A258286, A258287, A258288.

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

Adjacent sequences: A258277 A258278 A258279 * A258281 A258282 A258283

Keyword

nonn,tabf

Author

Alois P. Heinz, May 25 2015

Status

approved