A336133 Number of ways to split a strict integer partition of n into contiguous subsequences with strictly increasing sums.

1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 11, 14, 17, 22, 26, 35, 40, 51, 60, 75, 86, 109, 124, 153, 175, 214, 243, 297, 336, 403, 456, 546, 614, 731, 821, 975, 1095, 1283, 1437, 1689, 1887, 2195, 2448, 2851, 3172, 3676, 4083, 4724, 5245, 6022, 6677, 7695, 8504, 9720

Offset

0,4

Links

Table of n, a(n) for n=0..53.

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Example

The a(1) = 1 through a(9) = 9 splittings:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)    (5,3)    (5,4)
                          (4,1)  (5,1)    (5,2)    (6,2)    (6,3)
                                 (3,2,1)  (6,1)    (7,1)    (7,2)
                                          (4,2,1)  (4,3,1)  (8,1)
                                                   (5,2,1)  (4,3,2)
                                                            (5,3,1)
                                                            (6,2,1)
                                                            (4),(3,2)
The first splitting with more than two blocks is (8),(7,6),(5,4,3,2) under n = 35.

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, Select[IntegerPartitions[n], UnsameQ@@#&]}], {n, 0, 30}]

Crossrefs

The version with equal sums is A318683.

The version with strictly decreasing sums is A318684.

The version with weakly decreasing sums is A319794.

The version with different sums is A336132.

Starting with a composition gives A304961.

Starting with a non-strict partition gives A336134.

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, A316245, A317715, A323433, A336127, A336128, A336130, A336135.

Sequence in context: A008925 A266749 A308283 * A238625 A274145 A036072

Adjacent sequences: A336130 A336131 A336132 * A336134 A336135 A336136

Keyword

nonn

Author

Gus Wiseman, Jul 11 2020

Status

approved