A336134 Number of ways to split an integer partition of n into contiguous subsequences with strictly increasing sums.

1, 1, 2, 4, 6, 11, 17, 27, 37, 62, 82, 125, 168, 246, 320, 462, 585, 839, 1078, 1466, 1830, 2528, 3136, 4188, 5210, 6907, 8498, 11177, 13570, 17668, 21614, 27580, 33339, 42817, 51469, 65083, 78457, 98409, 117602, 147106, 174663, 217400, 259318, 319076, 377707

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..75

Example

The a(1) = 1 through a(6) = 17 splits:
  (1)  (2)    (3)        (4)          (5)            (6)
       (1,1)  (2,1)      (2,2)        (3,2)          (3,3)
              (1,1,1)    (3,1)        (4,1)          (4,2)
              (1),(1,1)  (2,1,1)      (2,2,1)        (5,1)
                         (1,1,1,1)    (3,1,1)        (2,2,2)
                         (1),(1,1,1)  (2,1,1,1)      (3,2,1)
                                      (2),(2,1)      (4,1,1)
                                      (1,1,1,1,1)    (2,2,1,1)
                                      (2),(1,1,1)    (2),(2,2)
                                      (1),(1,1,1,1)  (3,1,1,1)
                                      (1,1),(1,1,1)  (2,1,1,1,1)
                                                     (2),(2,1,1)
                                                     (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)

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], Less@@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+1, 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, 0)} \\ Andrew Howroyd, Jan 18 2024

Crossrefs

The version with equal sums is A317715.

The version with strictly decreasing sums is A336135.

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 A336133.

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, A318684, A323433, A336128, A336130.

Sequence in context: A294811 A333162 A336307 * A255214 A222047 A210520

Adjacent sequences: A336131 A336132 A336133 * A336135 A336136 A336137

Keyword

nonn

Author

Gus Wiseman, Jul 11 2020

Extensions

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

Status

approved