cp's OEIS Frontend

A108244 Triangle read by rows: row n gives list of all compositions of n ordered first by decreasing length, then by reverse colexicographical order.

%I A108244 #20 Jul 01 2025 01:02:00
%S A108244 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 A108244 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 A108244 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 A108244 Triangle read by rows: row n gives list of all compositions of n ordered first by decreasing length, then by reverse colexicographical order.
%C A108244 An example of a sequence which contains all finite sequences of positive integers as subsequences.
%C A108244 From _Andrey Zabolotskiy_, May 18 2018: (Start)
%C A108244 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 A108244 Length of k-th composition is A124748(k-1)+1.
%C A108244 Reversing every composition gives A296772. (End)
%H A108244 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Composition.html">Combinatorial composition</a>
%e A108244 The first 5 rows are:
%e A108244 {1}
%e A108244 {1, 1}, {2}
%e A108244 {1, 1, 1}, {1, 2}, {2, 1}, {3}
%e A108244 {1, 1, 1, 1}, {1, 1, 2}, {1, 2, 1}, {2, 1, 1}, {1, 3}, {2, 2}, {3, 1}, {4}
%e A108244 {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 A108244 Flatten[ Table[ Reverse[ # ] & /@ Reverse[ Sort[ Flatten[ Permutations[ # ] & /@ Partitions[ n], 1]]], {n, 6}]] (* _Robert G. Wilson v_, Jun 22 2005 *)
%Y A108244 Cf. A045623, A124748.
%Y A108244 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 A108244 nonn,tabf
%O A108244 1,4
%A A108244 _Hugo van der Sanden_, Jun 20 2005
%E A108244 More terms from _Robert G. Wilson v_, Jun 22 2005
%E A108244 Name corrected by _Andrey Zabolotskiy_, May 18 2018