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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. 12
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified July 19 10:32 EDT 2024. Contains 374392 sequences. (Running on oeis4.)