OFFSET
0,3
COMMENTS
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.
LINKS
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], GreaterEqual@@First/@Split[#, GreaterEqual]&]], {n, 0, 15}]
CROSSREFS
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 increasing leaders we have opposite A374634.
- For strictly decreasing leaders we have A374746.
A011782 counts compositions.
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.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 26 2024
STATUS
approved