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A124734 Table with all compositions sorted first by total, then by length and finally lexicographically.

%I A124734 #16 Jun 14 2020 09:44:54
%S A124734 1,2,1,1,3,1,2,2,1,1,1,1,4,1,3,2,2,3,1,1,1,2,1,2,1,2,1,1,1,1,1,1,5,1,
%T A124734 4,2,3,3,2,4,1,1,1,3,1,2,2,1,3,1,2,1,2,2,2,1,3,1,1,1,1,1,2,1,1,2,1,1,
%U A124734 2,1,1,2,1,1,1,1,1,1,1,1,6,1,5,2,4,3,3,4,2,5,1,1,1,4,1,2,3,1,3,2,1,4,1,2,1
%N A124734 Table with all compositions sorted first by total, then by length and finally lexicographically.
%C A124734 This is similar to the Abramowitz and Stegun ordering for partitions (see A036036). The standard ordering for compositions is A066099, which is more similar to the Mathematica partition ordering (A080577).
%C A124734 This can be regarded as a table in two ways: with each composition as a row, or with the compositions of each integer as a row. The first way has A124736 as row lengths and A070939 as row sums; the second has A001792 as row lengths and A001788 as row sums.
%C A124734 This sequence includes every finite sequence of positive integers.
%H A124734 Alois P. Heinz, <a href="/A124734/b124734.txt">Rows n = 1..11, flattened</a>
%H A124734 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 A124734 The table starts:
%e A124734 1
%e A124734 2; 1 1
%e A124734 3; 1 2; 2 1; 1 1 1
%e A124734 4; 1 3; 2 2; 3 1; 1 1 2; 1 2 1; 2 1 1; 1 1 1 1;
%t A124734 Table[Sort@Flatten[Permutations /@ IntegerPartitions@n, 1], {n, 8}] // Flatten (* _Robert Price_, Jun 13 2020 *)
%Y A124734 Cf. A001788, A001792, A036036, A066099, A070939, A080577, A124735, A124736.
%K A124734 easy,nonn,tabf
%O A124734 1,2
%A A124734 _Franklin T. Adams-Watters_, Nov 06 2006