A336136 Number of ways to split an integer partition of n into contiguous subsequences with weakly increasing sums.
1, 1, 3, 5, 11, 15, 31, 40, 73, 98, 158, 204, 340, 420, 629, 819, 1202, 1494, 2174, 2665, 3759, 4688, 6349, 7806, 10788, 13035, 17244, 21128, 27750, 33499, 43941, 52627, 67957, 81773, 103658, 124047, 158628, 187788, 235162, 280188, 349612, 413120, 513952, 604568
Offset: 0
Keywords
Examples
The a(1) = 1 through a(5) = 15 splittings:
(1) (2) (3) (4) (5)
(1,1) (2,1) (2,2) (3,2)
(1),(1) (1,1,1) (3,1) (4,1)
(1),(1,1) (2,1,1) (2,2,1)
(1),(1),(1) (2),(2) (3,1,1)
(1,1,1,1) (2,1,1,1)
(2),(1,1) (2),(2,1)
(1),(1,1,1) (1,1,1,1,1)
(1,1),(1,1) (2),(1,1,1)
(1),(1),(1,1) (1),(1,1,1,1)
(1),(1),(1),(1) (1,1),(1,1,1)
(1),(1),(1,1,1)
(1),(1,1),(1,1)
(1),(1),(1),(1,1)
(1),(1),(1),(1),(1)
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..60
Crossrefs
The version with weakly decreasing sums is A316245.
The version with equal sums is A317715.
The version with strictly increasing sums is A336134.
The version with strictly decreasing sums is A336135.
The version with different sums is A336131.
Starting with a composition gives A075900.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Programs
-
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],LessEqual@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}] -
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
Extensions
a(21) onwards from Andrew Howroyd, Jan 18 2024