OFFSET
0,4
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.
Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the minima are strictly decreasing.
LINKS
Christian Sievers, Table of n, a(n) for n = 0..500
EXAMPLE
The a(0) = 1 through a(9) = 16 compositions:
() (1) (2) (3) (4) (5) (6) (7) (8) (9)
(12) (13) (14) (15) (16) (17) (18)
(23) (24) (25) (26) (27)
(122) (123) (34) (35) (36)
(132) (124) (125) (45)
(133) (134) (126)
(142) (143) (135)
(152) (144)
(233) (153)
(1223) (162)
(1232) (234)
(243)
(1224)
(1233)
(1242)
(1323)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Less@@First/@Split[#, Less]&]], {n, 0, 15}]
CROSSREFS
The weak version is A374635.
The opposite version is A374763.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have A000041.
- For leaders of anti-runs we have A374679.
- For leaders of weakly increasing runs we have A374634.
- For leaders of strictly decreasing runs we have A374762.
Types of run-leaders (instead of strictly increasing):
- For strictly decreasing leaders we have A374689.
- For weakly increasing leaders we have A374690.
- For weakly decreasing leaders we have A374697.
A011782 counts 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