A336135 Number of ways to split an integer partition of n into contiguous subsequences with strictly decreasing sums.

1, 1, 2, 5, 8, 16, 29, 50, 79, 135, 213, 337, 522, 796, 1191, 1791, 2603, 3799, 5506, 7873, 11154, 15768, 21986, 30565, 42218, 57917, 78968, 107399, 144932, 194889, 261061, 347773, 461249, 610059, 802778, 1053173, 1377325, 1793985, 2329009, 3015922, 3891142

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

Example

The a(1) = 1 through a(5) = 16 splittings:
  (1)  (2)    (3)        (4)          (5)
       (1,1)  (2,1)      (2,2)        (3,2)
              (1,1,1)    (3,1)        (4,1)
              (2),(1)    (2,1,1)      (2,2,1)
              (1,1),(1)  (3),(1)      (3,1,1)
                         (1,1,1,1)    (3),(2)
                         (2,1),(1)    (4),(1)
                         (1,1,1),(1)  (2,1,1,1)
                                      (2,2),(1)
                                      (3),(1,1)
                                      (3,1),(1)
                                      (1,1,1,1,1)
                                      (2,1),(1,1)
                                      (2,1,1),(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], Greater@@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, 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

Crossrefs

The version with equal sums is A317715.

The version with strictly increasing sums is A336134.

The version with weakly increasing sums is A336136.

The version with weakly decreasing sums is A316245.

The version with different sums is A336131.

Starting with a composition gives A304961.

Starting with a strict partition gives A318684.

Partitions of partitions are A001970.

Partitions of compositions are A075900.

Compositions of compositions are A133494.

Compositions of partitions are A323583.

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

Sequence in context: A026007 A032233 A026530 * A032254 A323586 A048237

Adjacent sequences: A336132 A336133 A336134 * A336136 A336137 A336138

Keyword

nonn

Author

Gus Wiseman, Jul 11 2020

Extensions

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

Status

approved