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

Author

Alois P. Heinz, May 25 2015

Keywords

• nonn
• tabf

Examples

			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;
       ...
		

Links

• Alois P. Heinz, Rows n = 0..37, flattened

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.

Programs

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 *)

Formula

T(n,k) = 1/k! * [Product_{i=1..k} x_i^n] Product_{j>0} (1+Sum_{i=1..k} x_i^j).