OFFSET
0,4
COMMENTS
The leaders of increasing runs of a sequence are obtained by splitting it into maximal increasing subsequences and taking the first term of each.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
EXAMPLE
The a(1) = 1 through a(9) = 11 compositions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8)
(2,3) (2,4) (2,5) (2,6) (2,7)
(1,2,3) (3,4) (3,5) (3,6)
(1,3,2) (1,2,4) (1,2,5) (4,5)
(1,4,2) (1,3,4) (1,2,6)
(1,4,3) (1,3,5)
(1,5,2) (1,5,3)
(1,6,2)
(2,3,4)
(2,4,3)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], UnsameQ@@#&&Less@@First/@Split[#, Less]&]], {n, 0, 15}]
PROG
(PARI) \\ here Q(n) gives n-th row of A008289.
Q(n)={Vecrev(polcoef(prod(k=1, n, 1 + y*x^k, 1 + O(x*x^n)), n)/y)}
a(n)={if(n==0, 1, my(r=Q(n), s=Vec(serlaplace(exp(exp(x+O(x^#r))- 1)))); sum(k=1, #r, r[k]*s[k]))} \\ Andrew Howroyd, Sep 18 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 18 2024
EXTENSIONS
a(26) onwards from Andrew Howroyd, Sep 18 2024
STATUS
approved