A353859 - OEIS

%I #6 Jun 04 2022 22:19:18

%S 1,0,1,0,1,1,0,3,1,0,0,4,2,2,0,0,7,7,2,0,0,0,14,14,4,0,0,0,0,23,29,12,

%T 0,0,0,0,0,39,56,25,8,0,0,0,0,0,71,122,53,10,0,0,0,0,0,0,124,246,126,

%U 16,0,0,0,0,0,0,0,214,498,264,48,0,0,0,0,0,0,0

%N Triangle read by rows where T(n,k) is the number of integer compositions of n with composition run-sum trajectory of length k.

%C 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 (2,4,2,1,1) -> (2,4,2,2) -> (2,4,4) -> (2,8) is counted under T(10,4).

%e Triangle begins:

%e    1

%e    0   1

%e    0   1   1

%e    0   3   1   0

%e    0   4   2   2   0

%e    0   7   7   2   0   0

%e    0  14  14   4   0   0   0

%e    0  23  29  12   0   0   0   0

%e    0  39  56  25   8   0   0   0   0

%e    0  71 122  53  10   0   0   0   0   0

%e    0 124 246 126  16   0   0   0   0   0   0

%e    0 214 498 264  48   0   0   0   0   0   0   0

%e For example, row n = 5 counts the following compositions:

%e   (5)    (113)    (1121)

%e   (14)   (122)    (1211)

%e   (23)   (221)

%e   (32)   (311)

%e   (41)   (1112)

%e   (131)  (2111)

%e   (212)  (11111)

%t rsc[y_]:=If[y=={},{},NestWhileList[Total/@Split[#]&,y,MatchQ[#,{___,x_,x_,___}]&]];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[rsc[#]]==k&]],{n,0,10},{k,0,n}]

%Y Column k = 1 is A003242, ranked by A333489, complement A261983.

%Y Row sums are A011782.

%Y Positive row-lengths are A070939.

%Y The version for partitions is A353846, ranked by A353841.

%Y This statistic (trajectory length) is ranked by A353854, firsts A072639.

%Y Counting by length of last part instead of number of parts gives A353856.

%Y A333627 ranks the run-lengths of standard compositions.

%Y A353847 represents the run-sums of a composition, partitions A353832.

%Y A353853-A353859 pertain to composition run-sum trajectory.

%Y A353932 lists run-sums of standard compositions.

%Y Cf. A237685, A238279, A304442, A304465, A318928, A325277, A333755, A353848, A353850, A353852, A353855, A353858.

%K nonn,tabl

%O 0,8

%A _Gus Wiseman_, Jun 02 2022