OFFSET
0,3
COMMENTS
The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.
LINKS
Christian Sievers, Table of n, a(n) for n = 0..500
EXAMPLE
The composition (1,1,3,2,3,2) has strictly increasing runs ((1),(1,3),(2,3),(2)), with leaders (1,1,2,2), so is counted under a(12).
The a(0) = 1 through a(6) = 19 compositions:
() (1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(111) (22) (23) (24)
(112) (113) (33)
(121) (122) (114)
(1111) (131) (123)
(1112) (132)
(1121) (141)
(1211) (222)
(11111) (1113)
(1122)
(1131)
(1212)
(1311)
(11112)
(11121)
(11211)
(12111)
(111111)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], LessEqual@@First/@Split[#, Less]&]], {n, 0, 15}]
CROSSREFS
Ranked by positions of weakly increasing rows in A374683.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have A000041.
- For leaders of anti-runs we have A374681.
- For leaders of weakly increasing runs we have A374635.
- For leaders of weakly decreasing runs we have A188900.
- For leaders of strictly decreasing runs we have A374764.
Types of run-leaders (instead of weakly increasing):
- For strictly increasing leaders we have A374688.
- For strictly decreasing leaders we have A374689.
- For weakly decreasing leaders we have A374697.
A011782 counts compositions.
A335456 counts patterns matched by compositions.
A374700 counts compositions by sum of leaders of strictly increasing runs.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 27 2024
EXTENSIONS
a(26) and beyond from Christian Sievers, Aug 08 2024
STATUS
approved