OFFSET
0,3
COMMENTS
A composition of n is a finite sequence of positive integers summing to n.
LINKS
FORMULA
a(n) = Sum_{k=0..n} 2^(n-k) k! A008289(n,k).
EXAMPLE
The a(0) = 1 through a(4) = 16 splits:
() (1) (2) (3) (4)
(1,1) (1,2) (1,3)
(2,1) (2,2)
(1,1,1) (3,1)
(1),(2) (1,1,2)
(2),(1) (1,2,1)
(1),(1,1) (1),(3)
(1,1),(1) (2,1,1)
(3),(1)
(1,1,1,1)
(1),(1,2)
(1),(2,1)
(1,2),(1)
(2,1),(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], UnsameQ@@Total/@#&]], {ctn, Join@@Permutations/@IntegerPartitions[n]}], {n, 0, 10}]
CROSSREFS
The version with equal instead of different sums is A074854.
Starting with a strict composition gives A336128.
Starting with a partition gives A336131.
Starting with a strict partition gives A336132
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 09 2020
STATUS
approved