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 (1,3,1,4,1,2,2,1) has maximal weakly decreasing subsequences ((1),(3,1),(4,1),(2,2,1)), with leaders (1,3,4,2), so is counted under a(15).
The a(0) = 1 through a(5) = 15 compositions:
() (1) (2) (3) (4) (5)
(11) (12) (13) (14)
(21) (22) (23)
(111) (31) (32)
(112) (41)
(121) (113)
(211) (122)
(1111) (131)
(221)
(311)
(1112)
(1121)
(1211)
(2111)
(11111)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], UnsameQ@@First/@Split[#, GreaterEqual]&]], {n, 0, 15}]
CROSSREFS
Types of runs (instead of weakly decreasing):
Types of run-leaders (instead of distinct):
- For weakly increasing leaders we appear to have A188900.
- For identical leaders we have A374742.
- For strictly increasing leaders we have opposite A374634.
- For strictly decreasing leaders we have A374746.
- For weakly decreasing leaders we have A374747.
A011782 counts compositions.
A335456 counts patterns matched by compositions.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 25 2024
EXTENSIONS
a(24)-a(36) from Alois P. Heinz, Jul 26 2024
STATUS
approved