%I #20 Oct 21 2022 06:37:48
%S 1,2,1,2,3,1,3,4,1,4,2,3,5,1,2,3,1,5,2,4,6,1,2,4,1,6,2,5,3,4,7,1,2,5,
%T 1,3,4,1,7,2,6,3,5,8,1,2,6,1,3,5,1,8,2,3,4,2,7,3,6,4,5,9,1,2,3,4,1,2,
%U 7,1,3,6,1,4,5,1,9,2,3,5,2,8,3,7,4,6,10
%N Triangle in which n-th row lists lexicographically ordered increasing lists of parts of all partitions of n into distinct parts.
%H Alois P. Heinz, <a href="/A246688/b246688.txt">Rows n = 1..32, flattened</a>
%e Triangle begins:
%e [1];
%e [2];
%e [1,2], [3];
%e [1,3], [4];
%e [1,4], [2,3], [5];
%e [1,2,3], [1,5], [2,4], [6];
%e [1,2,4], [1,6], [2,5], [3,4], [7];
%e [1,2,5], [1,3,4], [1,7], [2,6], [3,5], [8];
%e [1,2,6], [1,3,5], [1,8], [2,3,4], [2,7], [3,6], [4,5], [9];
%e [1,2,3,4], [1,2,7], [1,3,6], [1,4,5], [1,9], [2,3,5], [2,8], [3,7], [4,6], [10];
%p b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
%p [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
%p end:
%p T:= n-> map(x-> x[], b(n, 1))[]:
%p seq(T(n), n=1..12);
%t T[n_] := Module[{ip, lg}, ip = Reverse /@ Select[ IntegerPartitions[n], # == DeleteDuplicates[#]&]; lg = Length /@ ip // Max; SortBy[PadRight[#, lg]&][ip]];
%t Table[T[n], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Oct 21 2022 *)
%Y Row lengths are A015723.
%Y Row sums give A066189.
%Y Last elements of rows are A000027.
%Y Cf. A026791, A026793, A118457, A265146.
%K nonn,tabf
%O 1,2
%A _Alois P. Heinz_, Sep 01 2014