A384391 -id:A384391 - cp's OEIS frontend

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

Gus Wiseman, May 28 2025

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.

			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)

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.

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}]

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

A384350 Number of subsets of {1..n} containing at least one element that is a sum of distinct non-elements.

Original entry on oeis.org

0, 0, 0, 1, 4, 13, 33, 81, 183, 402, 856, 1801, 3721, 7646, 15567, 31575

Offset: 0

Gus Wiseman, Jun 05 2025

Conjecture: Also the number of subsets of {1..n} such that it is possible in more than one way to choose a disjoint family of strict integer partitions, one of each element.

			For the set s = {1,5} we have 5 = 2+3, so s is counted under a(5).
The a(0) = 0 through a(5) = 13 subsets:
  .  .  .  {3}  {3}    {3}
                {4}    {4}
                {2,4}  {5}
                {3,4}  {1,5}
                       {2,4}
                       {2,5}
                       {3,4}
                       {3,5}
                       {4,5}
                       {1,4,5}
                       {2,3,5}
                       {2,4,5}
                       {3,4,5}

The complement is counted by A326080, allowing repeats A326083.
For strict partitions of n instead of subsets of {1..n} we have A384318, ranks A384322.
First differences are A384391.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A179009 counts maximally refined strict partitions, ranks A383707.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
A383706 counts ways to choose disjoint strict partitions of prime indices, non-disjoint A357982, non-strict A299200.
Cf. A279375, A279790, A317141, A317142, A383708, A383710, A384317, A384319, A384320, A384321.

nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@nonsets[#]]!={}&]],{n,0,10}]

A384392 Number of integer partitions of n whose distinct parts are maximally refined.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 7, 10, 14, 20, 24, 33, 41, 55, 70, 88, 110, 140, 171, 214, 265, 324, 397, 485, 588, 711, 861, 1032, 1241, 1486, 1773

Offset: 0

Gus Wiseman, Jun 07 2025

Given any partition, the following are equivalent:
1) The distinct parts are maximally refined.
2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
3) No part is a sum of distinct non-parts.

			The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)   (21)   (22)    (32)     (222)     (322)      (332)
       (11)  (111)  (31)    (41)     (321)     (331)      (431)
                    (211)   (221)    (411)     (421)      (521)
                    (1111)  (311)    (2211)    (2221)     (2222)
                            (2111)   (3111)    (3211)     (3221)
                            (11111)  (21111)   (4111)     (3311)
                                     (111111)  (22111)    (4211)
                                               (31111)    (22211)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

The strict case is A179009, ranks A383707.
For subsets instead of partitions we have A326080, complement A384350.
These partitions are ranked by A384320, complement A384321.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
Cf. A179822, A279375, A279790, A299200, A317142, A326083, A357982, A383706, A383708, A383710, A384317, A384318, A384319, A384391.

nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@nonsets[#]]=={}&]],{n,0,15}]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A384350 Number of subsets of {1..n} containing at least one element that is a sum of distinct non-elements.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A384392 Number of integer partitions of n whose distinct parts are maximally refined.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica