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%I #5 Jul 11 2020 07:38:56
%S 1,1,1,2,2,3,4,5,6,9,11,14,17,22,26,35,40,51,60,75,86,109,124,153,175,
%T 214,243,297,336,403,456,546,614,731,821,975,1095,1283,1437,1689,1887,
%U 2195,2448,2851,3172,3676,4083,4724,5245,6022,6677,7695,8504,9720
%N Number of ways to split a strict integer partition of n into contiguous subsequences with strictly increasing sums.
%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(1) = 1 through a(9) = 9 splittings:
%e (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (2,1) (3,1) (3,2) (4,2) (4,3) (5,3) (5,4)
%e (4,1) (5,1) (5,2) (6,2) (6,3)
%e (3,2,1) (6,1) (7,1) (7,2)
%e (4,2,1) (4,3,1) (8,1)
%e (5,2,1) (4,3,2)
%e (5,3,1)
%e (6,2,1)
%e (4),(3,2)
%e The first splitting with more than two blocks is (8),(7,6),(5,4,3,2) under n = 35.
%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],Less@@Total/@#&]],{ctn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]
%Y The version with equal sums is A318683.
%Y The version with strictly decreasing sums is A318684.
%Y The version with weakly decreasing sums is A319794.
%Y The version with different sums is A336132.
%Y Starting with a composition gives A304961.
%Y Starting with a non-strict partition gives A336134.
%Y Partitions of partitions are A001970.
%Y Partitions of compositions are A075900.
%Y Compositions of compositions are A133494.
%Y Compositions of partitions are A323583.
%Y Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336127, A336128, A336130, A336135.
%K nonn
%O 0,4
%A _Gus Wiseman_, Jul 11 2020
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