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A353859 Triangle read by rows where T(n,k) is the number of integer compositions of n with composition run-sum trajectory of length k.

%I A353859 #6 Jun 04 2022 22:19:18
%S A353859 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 A353859 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 A353859 16,0,0,0,0,0,0,0,214,498,264,48,0,0,0,0,0,0,0
%N A353859 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 A353859 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 A353859 Triangle begins:
%e A353859    1
%e A353859    0   1
%e A353859    0   1   1
%e A353859    0   3   1   0
%e A353859    0   4   2   2   0
%e A353859    0   7   7   2   0   0
%e A353859    0  14  14   4   0   0   0
%e A353859    0  23  29  12   0   0   0   0
%e A353859    0  39  56  25   8   0   0   0   0
%e A353859    0  71 122  53  10   0   0   0   0   0
%e A353859    0 124 246 126  16   0   0   0   0   0   0
%e A353859    0 214 498 264  48   0   0   0   0   0   0   0
%e A353859 For example, row n = 5 counts the following compositions:
%e A353859   (5)    (113)    (1121)
%e A353859   (14)   (122)    (1211)
%e A353859   (23)   (221)
%e A353859   (32)   (311)
%e A353859   (41)   (1112)
%e A353859   (131)  (2111)
%e A353859   (212)  (11111)
%t A353859 rsc[y_]:=If[y=={},{},NestWhileList[Total/@Split[#]&,y,MatchQ[#,{___,x_,x_,___}]&]];
%t A353859 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[rsc[#]]==k&]],{n,0,10},{k,0,n}]
%Y A353859 Column k = 1 is A003242, ranked by A333489, complement A261983.
%Y A353859 Row sums are A011782.
%Y A353859 Positive row-lengths are A070939.
%Y A353859 The version for partitions is A353846, ranked by A353841.
%Y A353859 This statistic (trajectory length) is ranked by A353854, firsts A072639.
%Y A353859 Counting by length of last part instead of number of parts gives A353856.
%Y A353859 A333627 ranks the run-lengths of standard compositions.
%Y A353859 A353847 represents the run-sums of a composition, partitions A353832.
%Y A353859 A353853-A353859 pertain to composition run-sum trajectory.
%Y A353859 A353932 lists run-sums of standard compositions.
%Y A353859 Cf. A237685, A238279, A304442, A304465, A318928, A325277, A333755, A353848, A353850, A353852, A353855, A353858.
%K A353859 nonn,tabl
%O A353859 0,8
%A A353859 _Gus Wiseman_, Jun 02 2022