A108244 - OEIS

%I #17 May 18 2018 19:49:20

%S 1,1,1,2,1,1,1,1,2,2,1,3,1,1,1,1,1,1,2,1,2,1,2,1,1,1,3,2,2,3,1,4,1,1,

%T 1,1,1,1,1,1,2,1,1,2,1,1,2,1,1,2,1,1,1,1,1,3,1,2,2,2,1,2,1,3,1,2,2,1,

%U 3,1,1,1,4,2,3,3,2,4,1,5,1,1,1,1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1

%N Triangle read by rows: row n gives list of all compositions of n ordered first by decreasing length, then by reverse colexicographical order.

%C An example of a sequence which contains all finite sequences of positive integers as subsequences.

%C From _Andrey Zabolotskiy_, May 18 2018: (Start)

%C At first, the ordering within the compositions of fixed length coincides with the lexicographical order (which is the case of A228369), but for n = 5 the partitions {2, 1, 2}, {1, 3, 1}, {2, 2, 1} go in this order because the order becomes reverse lexicographical when they are reversed (read right-to-left): {2, 1, 2}, {1, 3, 1}, {1, 2, 2}.

%C Length of k-th composition is A124748(k-1)+1.

%C Reversing every composition gives A296772. (End)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Composition.html">Combinatorial composition</a>

%e The first 5 rows are:

%e {1}

%e {1, 1}, {2}

%e {1, 1, 1}, {1, 2}, {2, 1}, {3}

%e {1, 1, 1, 1}, {1, 1, 2}, {1, 2, 1}, {2, 1, 1}, {1, 3}, {2, 2}, {3, 1}, {4}

%e {1, 1, 1, 1, 1}, {1, 1, 1, 2}, {1, 1, 2, 1}, {1, 2, 1, 1}, {2, 1, 1, 1}, {1, 1, 3}, {1, 2, 2}, {2, 1, 2}, {1, 3, 1}, {2, 2, 1}, {3, 1, 1}, {1, 4}, {2, 3}, {3, 2}, {4, 1}, {5}

%t Flatten[ Table[ Reverse[ # ] & /@ Reverse[ Sort[ Flatten[ Permutations[ # ] & /@ Partitions[ n], 1]]], {n, 6}]] (* _Robert G. Wilson v_, Jun 22 2005 *)

%Y Cf. A045623, A124748.

%Y Triangles of compositions: A066099 (main entry for compositions; similar to the Mathematica ordering for partitions, A080577), A124734 (similar to the Abramowitz & Stegun ordering for partitions, A036036), and this sequence (similar to the Maple partition ordering, A080576), A296772.

%K nonn,tabf

%O 1,4

%A _Hugo van der Sanden_, Jun 20, 2005

%E More terms from _Robert G. Wilson v_, Jun 22 2005

%E Name corrected by _Andrey Zabolotskiy_, May 18 2018