OFFSET
0,8
COMMENTS
The leaders of maximal weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
Also the number of integer compositions of n matching the dashed pattern 1-32, ranked by A375137.
Also the number of integer compositions of n matching the dashed pattern 23-1, ranked by A375138.
LINKS
EXAMPLE
- The maximal weakly increasing runs of y = (1,1,3,2,1) are ((1,1,3),(2),(1)) with leaders (1,2,1) so y is counted under a(8). Also, y matches 1-32 and avoids 23-1.
- The maximal weakly increasing runs of y = (1,3,2,1,1) are ((1,3),(2),(1,1)) with leaders (1,2,1) so y is counted under a(8). Also, y matches 1-32 and avoids 23-1.
- The maximal weakly increasing runs of y = (2,3,1,1,1) are ((2,3),(1,1,1)) with leaders (2,1) so y is not counted under a(8). Also, y avoids 1-32 and matches 23-1.
- The maximal weakly increasing runs of y = (2,3,2,1) are ((2,3),(2),(1)) with leaders (2,2,1) so y is not counted under a(8). Also, y avoids 1-32 and matches 23-1.
- The maximal weakly increasing runs of y = (2,1,3,1,1) are ((2),(1,3),(1,1)) with leaders (2,1,1) so y is not counted under a(8). Also, y avoids both 1-32 and 23-1.
- The maximal weakly increasing runs of y = (2,1,1,3,1) are ((2),(1,1,3),(1)) with leaders (2,1,1) so y is not counted under a(8). Also, y avoids both 1-32 and 23-1.
The a(0) = 0 through a(8) = 10 compositions:
. . . . . . (132) (142) (143)
(1132) (152)
(1321) (1142)
(1232)
(1322)
(1421)
(2132)
(11132)
(11321)
(13211)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], !GreaterEqual@@First/@Split[#, LessEqual]&]], {n, 0, 15}]
- or -
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#, {___, y_, z_, ___, x_, ___}/; x<y<z]&]], {n, 0, 15}]
CROSSREFS
The reverse version is the same.
For leaders of identical runs we have A056823.
The complement is counted by A189076.
The non-dashed version is A335514.
For weakly decreasing runs we have the complement of A374747.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 09 2024
STATUS
approved