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.
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 weakly decreasing [weakly increasing works too].
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1/(Product_{k>=1} (1 - x^k*Product_{j>=k+1} (1 + x^j))). - Andrew Howroyd, Jul 31 2024
EXAMPLE
The composition (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), so is not counted under a(12).
The a(0) = 1 through a(5) = 15 compositions:
() (1) (2) (3) (4) (5)
(11) (12) (13) (14)
(21) (22) (23)
(111) (31) (32)
(112) (41)
(121) (113)
(211) (131)
(1111) (212)
(221)
(311)
(1112)
(1121)
(1211)
(2111)
(11111)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], GreaterEqual@@First/@Split[#, Less]&]], {n, 0, 15}]
PROG
(PARI) seq(n) = Vec(1/prod(k=1, n, 1 - x^k*prod(j=k+1, n-k, 1 + x^j, 1 + O(x^(n-k+1))))) \\ Andrew Howroyd, Jul 31 2024
CROSSREFS
The opposite version is A374764.
Ranked by positions of weakly decreasing 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 A374682.
- For leaders of weakly decreasing runs we have A374747.
- For leaders of strictly decreasing runs we have A374765.
Types of run-leaders (instead of weakly decreasing):
- For weakly increasing leaders we have A374690.
- For strictly increasing leaders we have A374688.
- For strictly decreasing leaders we have A374689.
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 Andrew Howroyd, Jul 31 2024
STATUS
approved