This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A353856 #6 Jun 03 2022 07:42:43
%S A353856 1,0,1,0,2,0,0,2,2,0,0,5,2,1,0,0,2,12,2,0,0,0,8,10,12,2,0,0,0,2,32,23,
%T A353856 6,1,0,0,0,20,26,51,28,3,0,0,0,0,5,66,109,52,22,2,0,0,0,0,8,108,144,
%U A353856 188,53,10,1,0,0,0,0,2,134,358,282,196,48,4,0,0,0,0
%N A353856 Triangle read by rows where T(n,k) is the number of integer compositions of n with run-sum trajectory (condensation) ending in a composition of length k.
%C A353856 Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking the run-sums transformation (or condensation, represented by A353847) until an anti-run is reached. For example, the trajectory (2,1,1,3,1,1,2,1,1,2,1) -> (2,2,3,2,2,2,2,1) -> (4,3,8,1) is counted under T(15,4).
%e A353856 Triangle begins:
%e A353856 1
%e A353856 0 1
%e A353856 0 2 0
%e A353856 0 2 2 0
%e A353856 0 5 2 1 0
%e A353856 0 2 12 2 0 0
%e A353856 0 8 10 12 2 0 0
%e A353856 0 2 32 23 6 1 0 0
%e A353856 0 20 26 51 28 3 0 0 0
%e A353856 0 5 66 109 52 22 2 0 0 0
%e A353856 0 8 108 144 188 53 10 1 0 0 0
%e A353856 0 2 134 358 282 196 48 4 0 0 0 0
%e A353856 For example, row n = 6 counts the following compositions:
%e A353856 . (6) (15) (123) (1212) . .
%e A353856 (33) (24) (132) (2121)
%e A353856 (222) (42) (141)
%e A353856 (1113) (51) (213)
%e A353856 (2112) (114) (231)
%e A353856 (3111) (411) (312)
%e A353856 (11211) (1122) (321)
%e A353856 (111111) (2211) (1131)
%e A353856 (11112) (1221)
%e A353856 (21111) (1311)
%e A353856 (11121)
%e A353856 (12111)
%t A353856 Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],Length[FixedPoint[Total/@Split[#]&,#]]==k&]],{n,0,15},{k,0,n}]
%Y A353856 Row sums are A011782.
%Y A353856 Row-lengths without zeros appear to be A131737.
%Y A353856 The version for partitions is A353843.
%Y A353856 The length of the trajectory is A353854, firsts A072639, partitions A353841.
%Y A353856 The last part of the same trajectory is A353855.
%Y A353856 Column k = 1 is A353858.
%Y A353856 A066099 lists compositions in standard order.
%Y A353856 A318928 gives runs-resistance of binary expansion.
%Y A353856 A325268 counts partitions by omicron, rank statistic A304465.
%Y A353856 A333489 ranks anti-runs, counted by A003242 (complement A261983).
%Y A353856 A333627 ranks the run-lengths of standard compositions.
%Y A353856 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353856 A353847 represents the run-sums of a composition, partitions A353832.
%Y A353856 A353853-A353859 pertain to composition run-sum trajectory.
%Y A353856 A353932 lists run-sums of standard compositions.
%Y A353856 Cf. A237685, A238279, A304442, A325277, A333755, A353848, A353850, A353852.
%K A353856 nonn,tabl
%O A353856 0,5
%A A353856 _Gus Wiseman_, Jun 01 2022