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%I #30 Nov 01 2018 19:36:46
%S 1,2,3,5,6,7,10,11,14,15,13,21,22,30,17,26,33,35,42,19,34,39,55,66,70,
%T 23,38,51,65,77,78,105,110,29,46,57,85,91,102,130,154,165,210,31,58,
%U 69,95,114,119,143,170,182,195,231,330,37,62,87,115,133,138,187
%N Triangle T(n,k) in which n-th row lists in increasing order all partitions lambda of n into distinct parts encoded as Product_{i:lambda} prime(i); n>=0, 1<=k<=A000009(n).
%C The concatenation of all rows (with offset 1) gives a permutation of the squarefree numbers A005117. The missing positive numbers are in A013929.
%H Alois P. Heinz, <a href="/A246867/b246867.txt">Rows n = 0..42, flattened</a>
%e The partitions of n=5 into distinct parts are {[5], [4,1], [3,2]}, encodings give {prime(5), prime(4)*prime(1), prime(3)*prime(2)} = {11, 7*2, 5*3} => row 5 = [11, 14, 15].
%e For n=0 the empty partition [] gives the empty product 1.
%e Triangle T(n,k) begins:
%e 1;
%e 2;
%e 3;
%e 5, 6;
%e 7, 10;
%e 11, 14, 15;
%e 13, 21, 22, 30;
%e 17, 26, 33, 35, 42;
%e 19, 34, 39, 55, 66, 70;
%e 23, 38, 51, 65, 77, 78, 105, 110;
%e 29, 46, 57, 85, 91, 102, 130, 154, 165, 210;
%e ...
%e Corresponding triangle of strict integer partitions begins:
%e 0
%e (1)
%e (2)
%e (3) (21)
%e (4) (31)
%e (5) (41) (32)
%e (6) (42) (51) (321)
%e (7) (61) (52) (43) (421)
%e (8) (71) (62) (53) (521) (431)
%e (9) (81) (72) (63) (54) (621) (432) (531). - _Gus Wiseman_, Feb 23 2018
%p b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [], [seq(
%p map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..min(1, n/i))]))
%p end:
%p T:= n-> sort(b(n$2))[]:
%p seq(T(n), n=0..14);
%t b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Flatten[Table[Map[ #*Prime[i]^j&, b[n-i*j, i-1]], {j, 0, Min[1, n/i]}]]]]; T[n_] := Sort[b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 18 2016, after _Alois P. Heinz_ *)
%Y Column k=1 gives: A008578(n+1).
%Y Last elements of rows give: A246868.
%Y Row sums give A147655.
%Y Row lengths are: A000009.
%Y Cf. A005117, A118462, A215366 (the same for all partitions), A258323, A299755, A299757, A299759.
%K nonn,tabf,look
%O 0,2
%A _Alois P. Heinz_, Sep 05 2014
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