A246688 - OEIS

%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