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.
LINKS
EXAMPLE
The composition (3,1,2,1,1) has strictly decreasing runs ((3,1),(2,1),(1)), with leaders (3,2,1), so is counted under a(8).
The a(0) = 1 through a(8) = 15 compositions:
() (1) (2) (3) (4) (5) (6) (7) (8)
(21) (31) (32) (42) (43) (53)
(211) (41) (51) (52) (62)
(311) (312) (61) (71)
(321) (322) (413)
(411) (412) (422)
(421) (431)
(511) (512)
(3121) (521)
(3211) (611)
(3212)
(3221)
(4121)
(4211)
(31211)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Greater@@First/@Split[#, Greater]&]], {n, 0, 15}]
CROSSREFS
The opposite version is A374688.
The weak version is A374747.
For partitions instead of compositions we have A375133.
Other types of runs (instead of strictly decreasing):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we appear to have A188920.
- For leaders of anti-runs we have A374680.
- For leaders of strictly increasing runs we have A374689.
- For leaders of weakly decreasing runs we have A374746.
Other types of run-leaders (instead of strictly decreasing):
- For strictly increasing leaders we have A374762.
- For weakly increasing leaders we have A374764.
- For weakly decreasing leaders we have A374765.
A011782 counts compositions.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 30 2024
STATUS
approved