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A336127
Number of ways to split a composition of n into contiguous subsequences with different sums.
20
1, 1, 2, 8, 16, 48, 144, 352, 896, 2432, 7168, 16896, 46080, 114688, 303104, 843776, 2080768, 5308416, 13762560, 34865152, 87818240, 241172480, 583008256, 1503657984, 3762290688, 9604956160, 23689428992, 60532195328, 156397207552, 385137770496, 967978254336
OFFSET
0,3
COMMENTS
A composition of n is a finite sequence of positive integers summing to n.
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.
Sequence in context: A134353 A280229 A360323 * A076508 A162584 A100243
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 09 2020
STATUS
approved