A026793 - OEIS

%I #33 Jun 09 2021 11:15:01

%S 1,2,3,1,2,4,1,3,5,1,4,2,3,6,1,5,2,4,1,2,3,7,1,6,2,5,3,4,1,2,4,8,1,7,

%T 2,6,3,5,1,2,5,1,3,4,9,1,8,2,7,3,6,4,5,1,2,6,1,3,5,2,3,4,10,1,9,2,8,3,

%U 7,4,6,1,2,7,1,3,6,1,4,5,2,3,5,1,2,3,4,11,1,10,2,9,3,8,4,7,5,6,1,2,8,1,3,7,1,4,6,2,3,6,2,4

%N Juxtaposed partitions of 1,2,3,... into distinct parts, ordered by number of terms and then lexicographically.

%C This is the Abramowitz and Stegun ordering. - _Franklin T. Adams-Watters_, Apr 28 2006

%H Alois P. Heinz, <a href="/A026793/b026793.txt">Rows n = 1..32, flattened</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%e The partitions of 5 into distinct parts are [5], [1,4] and [2,3], so row 5 is 5,1,4,2,3.

%e Triangle begins:

%e [1];

%e [2];

%e [3], [1,2];

%e [4], [1,3];

%e [5], [1,4], [2,3];

%e [6], [1,5], [2,4], [1,2,3];

%e [7], [1,6], [2,5], [3,4], [1,2,4];

%e [8], [1,7], [2,6], [3,5], [1,2,5], [1,3,4];

%e [9], [1,8], [2,7], [3,6], [4,5], [1,2,6], [1,3,5], [2,3,4];

%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[], sort(b(n, 1)))[]:

%p seq(T(n), n=1..12);  # _Alois P. Heinz_, Jun 22 2020

%t Array[SortBy[Map[Reverse, Select[IntegerPartitions[#], UnsameQ @@ # &]], Length] &, 12] // Flatten (* _Michael De Vlieger_, Jun 22 2020 *)

%t b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i>n, {}, Join[Prepend[#, i]& /@ b[n-i, i+1], b[n, i+1]]]];

%t T[n_] := Sort[b[n, 1]];

%t Array[T, 12] // Flatten (* _Jean-François Alcover_, Jun 09 2021, after _Alois P. Heinz_ *)

%Y Cf. A118457, A118458 (partition lengths), A015723 (total row lengths), A036036, A000009, A246688.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_

%E Incorrect program removed by _Georg Fischer_, Jun 22 2020