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A374747
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Number of integer compositions of n whose leaders of weakly decreasing runs are themselves weakly decreasing.
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11
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1, 1, 2, 3, 5, 8, 14, 24, 43, 76, 136, 242, 431, 764, 1353, 2387, 4202, 7376, 12918, 22567, 39338, 68421, 118765, 205743
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OFFSET
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0,3
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COMMENTS
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The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.
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LINKS
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EXAMPLE
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The composition y = (3,2,1,2,2,1,2,5,1,1,1) has weakly decreasing runs ((3,2,1),(2,2,1),(2),(5,1,1,1)), with leaders (3,2,2,5), which are not weakly decreasing, so y is not counted under a(21).
The a(0) = 1 through a(6) = 14 compositions:
() (1) (2) (3) (4) (5) (6)
(11) (21) (22) (32) (33)
(111) (31) (41) (42)
(211) (212) (51)
(1111) (221) (222)
(311) (312)
(2111) (321)
(11111) (411)
(2112)
(2121)
(2211)
(3111)
(21111)
(111111)
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MATHEMATICA
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Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], GreaterEqual@@First/@Split[#, GreaterEqual]&]], {n, 0, 15}]
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CROSSREFS
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Ranked by positions of weakly decreasing rows in A374740, opposite A374629.
Types of runs (instead of weakly decreasing):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we appear to have A189076.
- For leaders of anti-runs we have A374682.
- For leaders of strictly increasing runs we have A374697.
- For leaders of strictly decreasing runs we have A374765.
Types of run-leaders (instead of weakly decreasing):
- For weakly increasing leaders we appear to have A188900.
- For strictly decreasing leaders we have A374746.
A124765 counts weakly decreasing runs in standard compositions.
A335456 counts patterns matched by compositions.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
Cf. A000009, A003242, A106356, A188920, A238343, A261982, A333213, A374630, A374635, A374636, A374741.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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