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%I #12 Feb 15 2024 01:15:22
%S 1,1,1,5,5,9,29,37,57,89,265,309,521,745,1129,3005,3545,5685,8201,
%T 12265,16629,41369,48109,77265,107645,160681,214861,316913,644837,
%U 798861,1207445,1694269,2437689,3326705,4710397,6270513,12246521,14853625,22244569,30308033,43706705,57926577,82166105,107873221,148081785,257989961,320873065,458994657,628016225,875485585,1165065733
%N Number of ways to split a strict composition of n into contiguous subsequences with different sums.
%C A composition of n is a finite sequence of positive integers summing to n.
%H Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>
%e The a(0) = 1 through a(5) = 5 splits:
%e () (1) (2) (3) (4) (5)
%e (12) (13) (14)
%e (21) (31) (23)
%e (1)(2) (1)(3) (32)
%e (2)(1) (3)(1) (41)
%e (1)(4)
%e (2)(3)
%e (3)(2)
%e (4)(1)
%e The a(6) = 29 splits:
%e (6) (1)(5) (1)(2)(3)
%e (15) (2)(4) (1)(3)(2)
%e (24) (4)(2) (2)(1)(3)
%e (42) (5)(1) (2)(3)(1)
%e (51) (1)(23) (3)(1)(2)
%e (123) (1)(32) (3)(2)(1)
%e (132) (13)(2)
%e (213) (2)(13)
%e (231) (2)(31)
%e (312) (23)(1)
%e (321) (31)(2)
%e (32)(1)
%t splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
%t Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,15}]
%Y The version with equal instead of different sums is A336130.
%Y Starting with a non-strict composition gives A336127.
%Y Starting with a partition gives A336131.
%Y Starting with a strict partition gives A336132.
%Y Partitions of partitions are A001970.
%Y Partitions of compositions are A075900.
%Y Compositions of compositions are A133494.
%Y Set partitions with distinct block-sums are A275780.
%Y Compositions of partitions are A323583.
%Y Cf. A006951, A063834, A271619, A279375, A305551, A326519, A317508, A318684, A336133, A336134, A336135.
%K nonn
%O 0,4
%A _Gus Wiseman_, Jul 10 2020
%E a(31)-a(50) from _Max Alekseyev_, Feb 14 2024
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