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A026793 Juxtaposed partitions of 1,2,3,... into distinct parts, ordered by number of terms and then lexicographically.

%I A026793 #33 Jun 09 2021 11:15:01
%S A026793 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 A026793 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 A026793 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 A026793 Juxtaposed partitions of 1,2,3,... into distinct parts, ordered by number of terms and then lexicographically.
%C A026793 This is the Abramowitz and Stegun ordering. - _Franklin T. Adams-Watters_, Apr 28 2006
%H A026793 Alois P. Heinz, <a href="/A026793/b026793.txt">Rows n = 1..32, flattened</a>
%H A026793 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 A026793 The partitions of 5 into distinct parts are [5], [1,4] and [2,3], so row 5 is 5,1,4,2,3.
%e A026793 Triangle begins:
%e A026793 [1];
%e A026793 [2];
%e A026793 [3], [1,2];
%e A026793 [4], [1,3];
%e A026793 [5], [1,4], [2,3];
%e A026793 [6], [1,5], [2,4], [1,2,3];
%e A026793 [7], [1,6], [2,5], [3,4], [1,2,4];
%e A026793 [8], [1,7], [2,6], [3,5], [1,2,5], [1,3,4];
%e A026793 [9], [1,8], [2,7], [3,6], [4,5], [1,2,6], [1,3,5], [2,3,4];
%p A026793 b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
%p A026793       [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
%p A026793     end:
%p A026793 T:= n-> map(x-> x[], sort(b(n, 1)))[]:
%p A026793 seq(T(n), n=1..12);  # _Alois P. Heinz_, Jun 22 2020
%t A026793 Array[SortBy[Map[Reverse, Select[IntegerPartitions[#], UnsameQ @@ # &]], Length] &, 12] // Flatten (* _Michael De Vlieger_, Jun 22 2020 *)
%t A026793 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 A026793 T[n_] := Sort[b[n, 1]];
%t A026793 Array[T, 12] // Flatten (* _Jean-François Alcover_, Jun 09 2021, after _Alois P. Heinz_ *)
%Y A026793 Cf. A118457, A118458 (partition lengths), A015723 (total row lengths), A036036, A000009, A246688.
%K A026793 nonn,tabf
%O A026793 1,2
%A A026793 _Clark Kimberling_
%E A026793 Incorrect program removed by _Georg Fischer_, Jun 22 2020