OFFSET
0,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
EXAMPLE
The a(1) = 1 through a(5) = 16 splittings:
(1) (2) (3) (4) (5)
(1,1) (2,1) (2,2) (3,2)
(1,1,1) (3,1) (4,1)
(2),(1) (2,1,1) (2,2,1)
(1,1),(1) (3),(1) (3,1,1)
(1,1,1,1) (3),(2)
(2,1),(1) (4),(1)
(1,1,1),(1) (2,1,1,1)
(2,2),(1)
(3),(1,1)
(3,1),(1)
(1,1,1,1,1)
(2,1),(1,1)
(2,1,1),(1)
(1,1,1),(1,1)
(1,1,1,1),(1)
MATHEMATICA
splits[dom_]:=Append[Join@@Table[Prepend[#, Take[dom, i]]&/@splits[Drop[dom, i]], {i, Length[dom]-1}], {dom}];
Table[Sum[Length[Select[splits[ctn], Greater@@Total/@#&]], {ctn, IntegerPartitions[n]}], {n, 0, 10}]
PROG
(PARI) a(n)={my(recurse(r, m, s, t, f)=if(m==0, r==0, if(f, self()(r, min(m, t-1), t-1, 0, 0)) + self()(r, m-1, s, t, 0) + if(t+m<=s, self()(r-m, min(m, r-m), s, t+m, 1)))); recurse(n, n, n, 0)} \\ Andrew Howroyd, Jan 18 2024
CROSSREFS
The version with equal sums is A317715.
The version with strictly increasing sums is A336134.
The version with weakly increasing sums is A336136.
The version with weakly decreasing sums is A316245.
The version with different sums is A336131.
Starting with a composition gives A304961.
Starting with a strict partition gives A318684.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 11 2020
EXTENSIONS
a(21) onwards from Andrew Howroyd, Jan 18 2024
STATUS
approved