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

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..60

Example

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)

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}]

Prog

(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

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.

Cf. A006951, A063834, A279786, A304961, A305551, A318684, A323433, A336128, A336130, A336133.

Sequence in context: A200176 A092929 A138879 * A318915 A322439 A018313

Adjacent sequences: A336133 A336134 A336135 * A336137 A336138 A336139

Keyword

nonn

Author

Gus Wiseman, Jul 11 2020

Extensions

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

Status

approved