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%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
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