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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.

%I A258280 #27 Feb 17 2021 14:07:58
%S A258280 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 A258280 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 A258280 103,75,29,7,1,1,22,95,154,118,47,10,1,1,27,140,279,266,134,40,8,1
%N 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.
%H A258280 Alois P. Heinz, <a href="/A258280/b258280.txt">Rows n = 0..37, flattened</a>
%F A258280 T(n,k) = 1/k! * [Product_{i=1..k} x_i^n] Product_{j>0} (1+Sum_{i=1..k} x_i^j).
%e A258280 T(7,0) = 1: [].
%e A258280 T(7,1) = 5: [7], [6,1], [5,2], [4,3], [4,2,1].
%e A258280 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 A258280 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 A258280 T(7,4) = 1: [7;6,1;5,2;4,3].
%e A258280 T(8,4) = 1: [8;7,1;6,2;5,3].
%e A258280 Triangle T(n,k) begins:
%e A258280 00  :  1,  1;
%e A258280 01  :  1,  1;
%e A258280 02  :  1,  1;
%e A258280 03  :  1,  2,   1;
%e A258280 04  :  1,  2,   1;
%e A258280 05  :  1,  3,   3,   1;
%e A258280 06  :  1,  4,   4,   1;
%e A258280 07  :  1,  5,   7,   4,   1;
%e A258280 08  :  1,  6,   9,   5,   1;
%e A258280 09  :  1,  8,  16,  13,   5,   1;
%e A258280 10  :  1, 10,  21,  18,   7,   1;
%e A258280 11  :  1, 12,  33,  37,  20,   6,  1;
%e A258280 12  :  1, 15,  46,  56,  31,   8,  1;
%e A258280 13  :  1, 18,  68, 103,  75,  29,  7, 1;
%e A258280 14  :  1, 22,  95, 154, 118,  47, 10, 1;
%e A258280 15  :  1, 27, 140, 279, 266, 134, 40, 8, 1;
%e A258280        ...
%p A258280 b:= proc() option remember; local m; m:= args[nargs];
%p A258280      `if`(nargs=1, 1, `if`(args[1]=0, b(args[t] $t=2..nargs),
%p A258280      `if`(m=0 or add(args[i], i=1..nargs-1)> m*(m+1)/2, 0,
%p A258280       b(args[t] $t=1..nargs-1, m-1) +add(`if`(args[j]-m<0, 0,
%p A258280       b(sort([seq(args[i] -`if`(i=j, m, 0), i=1..nargs-1)])[]
%p A258280       , m-1)), j=1..nargs-1))))
%p A258280     end:
%p A258280 T:= (n, k)-> b(n$k+1)/k!:
%p A258280 seq(seq(T(n, k), k=0..max(1, ceil(n/2))), n=0..15);
%t A258280 disParts[n_] := disParts[n] = Select[IntegerPartitions[n], Length[#] == Length[Union[#]]&];
%t A258280 T[n_, k_] := Select[Subsets[disParts[n], {k}], Length[Flatten[#]] == Length[Union[Flatten[#]]]&] // Length;
%t A258280 Table[T[n, k], {n, 0, 15}, {k, 0, Max[1, Ceiling[n/2]]}] // Flatten (* _Jean-François Alcover_, Feb 17 2021 *)
%Y A258280 Columns k=0-10 give: A000012, A000009, A108796, A258281, A258282, A258283, A258284, A258285, A258286, A258287, A258288.
%Y A258280 Row sums give 1 + A258289.
%Y A258280 Row lengths give 1 + A065033.
%Y A258280 T(n^2,n) gives A284824.
%K A258280 nonn,tabf
%O A258280 0,8
%A A258280 _Alois P. Heinz_, May 25 2015