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A374761
Number of integer compositions of n whose leaders of strictly decreasing runs are distinct.
30
1, 1, 1, 3, 5, 7, 13, 27, 45, 73, 117, 205, 365, 631, 1061, 1711, 2777, 4599, 7657, 12855, 21409, 35059, 56721, 91149
OFFSET
0,4
COMMENTS
The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.
EXAMPLE
The composition (3,1,4,3,2,1,2,8) has strictly decreasing runs ((3,1),(4,3,2,1),(2),(8)), with leaders (3,4,2,1), so is counted under a(24).
The a(0) = 1 through a(6) = 13 compositions:
() (1) (2) (3) (4) (5) (6)
(12) (13) (14) (15)
(21) (31) (23) (24)
(121) (32) (42)
(211) (41) (51)
(131) (123)
(311) (132)
(141)
(213)
(231)
(312)
(321)
(411)
MATHEMATICA
Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n], UnsameQ@@First/@Split[#, Greater]&]], {n, 0, 15}]
CROSSREFS
For leaders of identical runs we have A274174, ranked by A374249.
The weak opposite version is A374632, ranks A374768.
The opposite version is A374687, ranks A374698.
For identical instead of distinct leaders we have A374760, ranks A374759.
The weak version is A374743, ranks A374701.
Ranked by A374767.
For partitions instead of compositions we have A375133.
Other types of runs:
- For leaders of identical runs we have A000005 for n > 0, ranks A272919.
- For leaders of anti-runs we have A374518, ranked by A374638.
Other types of run-leaders:
- For strictly increasing leaders we have A374762.
- For strictly decreasing leaders we have A374763.
- For weakly increasing leaders we have A374764.
- For weakly decreasing leaders we have A374765.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Sequence in context: A024724 A024946 A257762 * A224222 A224221 A128547
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 29 2024
STATUS
approved