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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: Product_{i>0} (1 + (x^i)*Product_{j>i} (1 + x^j)). - John Tyler Rascoe, Jul 29 2024
EXAMPLE
The a(0) = 1 through a(8) = 21 compositions:
() (1) (2) (3) (4) (5) (6) (7) (8)
(12) (13) (14) (15) (16) (17)
(21) (31) (23) (24) (25) (26)
(32) (42) (34) (35)
(41) (51) (43) (53)
(212) (123) (52) (62)
(213) (61) (71)
(231) (124) (125)
(312) (214) (134)
(321) (241) (215)
(313) (251)
(412) (314)
(421) (323)
(341)
(413)
(431)
(512)
(521)
(2123)
(2312)
(3212)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Greater@@First/@Split[#, Less]&]], {n, 0, 15}]
PROG
(PARI)
C_x(N) = {my(x='x+O('x^N), h=prod(i=1, N, 1+(x^i)*prod(j=i+1, N, 1+x^j))); Vec(h)}
C_x(50) \\ John Tyler Rascoe, Jul 29 2024
CROSSREFS
The weak version appears to be A189076.
Ranked by positions of strictly decreasing rows in A374683.
The opposite version is A374762.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have A000041.
- For leaders of anti-runs we have A374680.
- For leaders of weakly increasing runs we have A188920.
- For leaders of weakly decreasing runs we have A374746.
- For leaders of strictly decreasing runs we have A374763.
Types of run-leaders (instead of strictly decreasing):
- For strictly increasing leaders we have A374688.
- For weakly increasing leaders we have A374690.
- 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) onwards from John Tyler Rascoe, Jul 29 2024
STATUS
approved