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A353859
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Triangle read by rows where T(n,k) is the number of integer compositions of n with composition run-sum trajectory of length k.
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33
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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, 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, 16, 0, 0, 0, 0, 0, 0, 0, 214, 498, 264, 48, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0,8
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COMMENTS
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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).
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LINKS
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EXAMPLE
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Triangle begins:
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 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 16 0 0 0 0 0 0
0 214 498 264 48 0 0 0 0 0 0 0
For example, row n = 5 counts the following compositions:
(5) (113) (1121)
(14) (122) (1211)
(23) (221)
(32) (311)
(41) (1112)
(131) (2111)
(212) (11111)
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MATHEMATICA
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rsc[y_]:=If[y=={}, {}, NestWhileList[Total/@Split[#]&, y, MatchQ[#, {___, x_, x_, ___}]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Length[rsc[#]]==k&]], {n, 0, 10}, {k, 0, n}]
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CROSSREFS
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This statistic (trajectory length) is ranked by A353854, firsts A072639.
Counting by length of last part instead of number of parts gives A353856.
A333627 ranks the run-lengths of standard compositions.
A353932 lists run-sums of standard compositions.
Cf. A237685, A238279, A304442, A304465, A318928, A325277, A333755, A353848, A353850, A353852, A353855, A353858.
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KEYWORD
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AUTHOR
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STATUS
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approved
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