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%I #11 Jan 19 2024 02:19:36
%S 1,1,2,5,8,16,29,50,79,135,213,337,522,796,1191,1791,2603,3799,5506,
%T 7873,11154,15768,21986,30565,42218,57917,78968,107399,144932,194889,
%U 261061,347773,461249,610059,802778,1053173,1377325,1793985,2329009,3015922,3891142
%N Number of ways to split an integer partition of n into contiguous subsequences with strictly decreasing sums.
%H Andrew Howroyd, <a href="/A336135/b336135.txt">Table of n, a(n) for n = 0..50</a>
%e The a(1) = 1 through a(5) = 16 splittings:
%e (1) (2) (3) (4) (5)
%e (1,1) (2,1) (2,2) (3,2)
%e (1,1,1) (3,1) (4,1)
%e (2),(1) (2,1,1) (2,2,1)
%e (1,1),(1) (3),(1) (3,1,1)
%e (1,1,1,1) (3),(2)
%e (2,1),(1) (4),(1)
%e (1,1,1),(1) (2,1,1,1)
%e (2,2),(1)
%e (3),(1,1)
%e (3,1),(1)
%e (1,1,1,1,1)
%e (2,1),(1,1)
%e (2,1,1),(1)
%e (1,1,1),(1,1)
%e (1,1,1,1),(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],Greater@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]
%o (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
%Y The version with equal sums is A317715.
%Y The version with strictly increasing sums is A336134.
%Y The version with weakly increasing sums is A336136.
%Y The version with weakly decreasing sums is A316245.
%Y The version with different sums is A336131.
%Y Starting with a composition gives A304961.
%Y Starting with a strict partition gives A318684.
%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, A323433, A336128, A336130, A336133.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jul 11 2020
%E a(21) onwards from _Andrew Howroyd_, Jan 18 2024