A316245 Number of ways to split an integer partition of n into consecutive subsequences with weakly decreasing sums.

1, 1, 3, 6, 14, 25, 52, 89, 167, 279, 486, 786, 1322, 2069, 3326, 5128, 8004, 12055, 18384, 27203, 40588, 59186, 86645, 124583, 179784, 255111, 362767, 509319, 715422, 993681, 1380793, 1899630, 2613064, 3564177, 4857631, 6572314, 8884973, 11930363, 16002853

Offset: 0

Author

Gus Wiseman, Sep 29 2018

Keywords

• nonn

Examples

			The a(4) = 14 split partitions:
  (4)
  (31)
  (22)
  (211)
  (3)(1)
  (2)(2)
  (1111)
  (21)(1)
  (2)(11)
  (111)(1)
  (11)(11)
  (2)(1)(1)
  (11)(1)(1)
  (1)(1)(1)(1)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50
• Gus Wiseman, The a(5) = 25 split partitions.
• Gus Wiseman, The a(6) = 52 split partitions.

Crossrefs

Cf. A000041, A001970, A063834, A316223, A317546, A319001, A319003.

Cf. A317508, A317715, A318434, A318683, A318684, A319794.

Programs

Mathematica

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Sum[Length[Select[comps[y],OrderedQ[Total/@#,GreaterEqual]&]],{y,IntegerPartitions[n]}],{n,10}]

PARI

a(n)={my(recurse(r,m,s,t,f)=if(m==0, r==0, if(f, self()(r,min(m,t),t,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,0,0)} \\ Andrew Howroyd, Jan 18 2024

Extensions

a(21) onwards from Andrew Howroyd, Jan 18 2024