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Number of ways to split an integer partition of n into contiguous subsequences with weakly increasing sums.
5

%I #10 Jan 19 2024 02:19:59

%S 1,1,3,5,11,15,31,40,73,98,158,204,340,420,629,819,1202,1494,2174,

%T 2665,3759,4688,6349,7806,10788,13035,17244,21128,27750,33499,43941,

%U 52627,67957,81773,103658,124047,158628,187788,235162,280188,349612,413120,513952,604568

%N Number of ways to split an integer partition of n into contiguous subsequences with weakly increasing sums.

%H Andrew Howroyd, <a href="/A336136/b336136.txt">Table of n, a(n) for n = 0..60</a>

%e The a(1) = 1 through a(5) = 15 splittings:

%e (1) (2) (3) (4) (5)

%e (1,1) (2,1) (2,2) (3,2)

%e (1),(1) (1,1,1) (3,1) (4,1)

%e (1),(1,1) (2,1,1) (2,2,1)

%e (1),(1),(1) (2),(2) (3,1,1)

%e (1,1,1,1) (2,1,1,1)

%e (2),(1,1) (2),(2,1)

%e (1),(1,1,1) (1,1,1,1,1)

%e (1,1),(1,1) (2),(1,1,1)

%e (1),(1),(1,1) (1),(1,1,1,1)

%e (1),(1),(1),(1) (1,1),(1,1,1)

%e (1),(1),(1,1,1)

%e (1),(1,1),(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],LessEqual@@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 && r >= t && t >= s, self()(r,m,t,0,0)) + self()(r,m-1,s,t,0) + self()(r-m,min(m,r-m),s,t+m,1))); recurse(n,n,0,0)} \\ _Andrew Howroyd_, Jan 18 2024

%Y The version with weakly decreasing sums is A316245.

%Y The version with equal sums is A317715.

%Y The version with strictly increasing sums is A336134.

%Y The version with strictly decreasing sums is A336135.

%Y The version with different sums is A336131.

%Y Starting with a composition gives A075900.

%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, A304961, A305551, A318684, 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