A384318 Number of strict integer partitions of n that are not maximally refined.

0, 0, 0, 1, 1, 1, 3, 4, 4, 5, 9, 10, 13, 15, 17, 26, 29, 36, 43, 49, 57, 74, 84, 101, 118, 136, 158, 181, 219, 248, 291

Offset

0,7

Comments

This is the number of strict integer partitions of n containing at least one sum of distinct non-parts.

Conjecture: Also the number of strict integer partitions of n such that it is possible in more than one way to choose a disjoint family of strict integer partitions, one of each part.

Links

Table of n, a(n) for n=0..30.

Formula

a(n) = A000009(n) - A179009(n).

Example

For y = (5,4,2) we have 4 = 3+1 so y is counted under a(11).
On the other hand, no part of z = (6,4,1) is a subset-sum of the non-parts {2,3,5}, so z is not counted under a(11).
The a(3) = 1 through a(11) = 10 strict partitions:
  (3)  (4)  (5)  (6)    (7)    (8)    (9)    (10)     (11)
                 (4,2)  (4,3)  (5,3)  (5,4)  (6,4)    (6,5)
                 (5,1)  (5,2)  (6,2)  (6,3)  (7,3)    (7,4)
                        (6,1)  (7,1)  (7,2)  (8,2)    (8,3)
                                      (8,1)  (9,1)    (9,2)
                                             (5,3,2)  (10,1)
                                             (5,4,1)  (5,4,2)
                                             (6,3,1)  (6,3,2)
                                             (7,2,1)  (7,3,1)
                                                      (8,2,1)

Mathematica

nonsets[y_]:=If[Length[y]==0, {}, Rest[Subsets[Complement[Range[Max@@y], y]]]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Intersection[#, Total/@nonsets[#]]!={}&]], {n, 0, 30}]

Crossrefs

The strict complement is A179009, ranks A383707.

The non-strict version for at least one choice is A383708, for none A383710.

The non-strict version is A384317, ranks A384321, complement A384392, ranks A384320.

These partitions are ranked by A384322.

For subsets instead of partitions we have A384350, complement A326080.

Cf. A357982, A383706 (disjoint), A384319, A384323 (non-strict).

Cf. A048767, A098859, A179822, A239455, A279375, A317142, A351293, A382525, A383533, A383711, A384391.

Sequence in context: A051665 A028263 A059179 * A279678 A381552 A222283

Adjacent sequences: A384315 A384316 A384317 * A384319 A384320 A384321

Keyword

nonn,more

Author

Gus Wiseman, May 28 2025

Status

approved