OFFSET
0,3
COMMENTS
The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1 + Sum_{k>=1} -1 + 1/(1 - x^k*Product_{j=1..k-1} (1 + x^j)). - Andrew Howroyd, Jul 31 2024
EXAMPLE
The composition (3,3,2,1,3,2,1) has strictly decreasing runs ((3),(3,2,1),(3,2,1)), with leaders (3,3,3), so is counted under a(15).
The a(0) = 1 through a(8) = 15 compositions:
() (1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (212) (51) (61) (62)
(221) (222) (313) (71)
(11111) (321) (331) (323)
(2121) (421) (332)
(111111) (2122) (431)
(2212) (521)
(2221) (2222)
(1111111) (3131)
(21212)
(21221)
(22121)
(11111111)
MATHEMATICA
Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n], SameQ@@First/@Split[#, Greater]&]], {n, 0, 15}]
PROG
(PARI) seq(n) = Vec(1 + sum(k=1, n, 1/(1 - x^k*prod(j=1, min(n-k, k-1), 1 + x^j, 1 + O(x^(n-k+1))))-1)) \\ Andrew Howroyd, Jul 31 2024
CROSSREFS
For partitions instead of compositions we have A034296.
Ranked by A374759.
Other types of runs (instead of strictly decreasing):
Other types of run-leaders (instead of identical):
- For strictly increasing leaders we have A374762.
- For strictly decreasing leaders we have A374763.
- For weakly increasing leaders we have A374764.
- For weakly decreasing leaders we have A374765.
A011782 counts compositions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 29 2024
EXTENSIONS
a(24) onwards from Andrew Howroyd, Jul 31 2024
STATUS
approved