A258280 - OEIS

%I #27 Feb 17 2021 14:07:58

%S 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,

%T 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,

%U 103,75,29,7,1,1,22,95,154,118,47,10,1,1,27,140,279,266,134,40,8,1

%N 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.

%H Alois P. Heinz, <a href="/A258280/b258280.txt">Rows n = 0..37, flattened</a>

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

%e T(7,0) = 1: [].

%e T(7,1) = 5: [7], [6,1], [5,2], [4,3], [4,2,1].

%e 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].

%e 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].

%e T(7,4) = 1: [7;6,1;5,2;4,3].

%e T(8,4) = 1: [8;7,1;6,2;5,3].

%e Triangle T(n,k) begins:

%e 00  :  1,  1;

%e 01  :  1,  1;

%e 02  :  1,  1;

%e 03  :  1,  2,   1;

%e 04  :  1,  2,   1;

%e 05  :  1,  3,   3,   1;

%e 06  :  1,  4,   4,   1;

%e 07  :  1,  5,   7,   4,   1;

%e 08  :  1,  6,   9,   5,   1;

%e 09  :  1,  8,  16,  13,   5,   1;

%e 10  :  1, 10,  21,  18,   7,   1;

%e 11  :  1, 12,  33,  37,  20,   6,  1;

%e 12  :  1, 15,  46,  56,  31,   8,  1;

%e 13  :  1, 18,  68, 103,  75,  29,  7, 1;

%e 14  :  1, 22,  95, 154, 118,  47, 10, 1;

%e 15  :  1, 27, 140, 279, 266, 134, 40, 8, 1;

%e        ...

%p b:= proc() option remember; local m; m:= args[nargs];

%p      `if`(nargs=1, 1, `if`(args[1]=0, b(args[t] $t=2..nargs),

%p      `if`(m=0 or add(args[i], i=1..nargs-1)> m*(m+1)/2, 0,

%p       b(args[t] $t=1..nargs-1, m-1) +add(`if`(args[j]-m<0, 0,

%p       b(sort([seq(args[i] -`if`(i=j, m, 0), i=1..nargs-1)])[]

%p       , m-1)), j=1..nargs-1))))

%p     end:

%p T:= (n, k)-> b(n$k+1)/k!:

%p seq(seq(T(n, k), k=0..max(1, ceil(n/2))), n=0..15);

%t disParts[n_] := disParts[n] = Select[IntegerPartitions[n], Length[#] == Length[Union[#]]&];

%t T[n_, k_] := Select[Subsets[disParts[n], {k}], Length[Flatten[#]] == Length[Union[Flatten[#]]]&] // Length;

%t Table[T[n, k], {n, 0, 15}, {k, 0, Max[1, Ceiling[n/2]]}] // Flatten (* _Jean-François Alcover_, Feb 17 2021 *)

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

%Y Row sums give 1 + A258289.

%Y Row lengths give 1 + A065033.

%Y T(n^2,n) gives A284824.

%K nonn,tabf

%O 0,8

%A _Alois P. Heinz_, May 25 2015