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/(Product_{k>=1} (1 - x^k*Product_{j=1..k-1} (1 + x^j))). - Andrew Howroyd, Jul 31 2024
EXAMPLE
The composition (1,1,2,1) has strictly decreasing runs ((1),(1),(2,1)) with leaders (1,1,2) so is counted under a(5).
The composition (1,2,1,1) has strictly decreasing runs ((1),(2,1),(1)) with leaders (1,2,1) so is not counted under a(5).
The a(0) = 1 through a(5) = 13 compositions:
() (1) (2) (3) (4) (5)
(11) (12) (13) (14)
(21) (22) (23)
(111) (31) (32)
(112) (41)
(121) (113)
(1111) (122)
(131)
(212)
(221)
(1112)
(1121)
(11111)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], LessEqual@@First/@Split[#, Greater]&]], {n, 0, 15}]
PROG
(PARI) seq(n) = Vec(1/prod(k=1, n, 1 - x^k*prod(j=1, min(n-k, k-1), 1 + x^j, 1 + O(x^(n-k+1))))) \\ Andrew Howroyd, Jul 31 2024
CROSSREFS
For partitions instead of compositions we have A034296.
For strictly increasing leaders we have A374688.
The opposite version is A374697.
Other types of runs (instead of strictly decreasing):
- 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 strictly increasing runs we have A374690.
- For leaders of weakly decreasing runs we have A188900.
Other types of run-leaders (instead of weakly increasing):
- For strictly increasing leaders we have A374762.
- For weakly decreasing leaders we have A374765.
- For strictly decreasing leaders we have A374763.
A011782 counts compositions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 30 2024
EXTENSIONS
a(24) onwards from Andrew Howroyd, Jul 31 2024
STATUS
approved