A384319 -id:A384319 - cp's OEIS frontend

A384391 Number of subsets of {1..n} containing n and some element that is a sum of distinct non-elements.

0, 0, 1, 3, 9, 20, 48, 102, 219, 454, 945, 1920, 3925, 7921, 16008

Offset: 0

Gus Wiseman, Jun 06 2025

			The a(0) = 0 through a(6) = 20 subsets:
  .  .  .  {3}  {4}    {5}      {6}
                {2,4}  {1,5}    {1,6}
                {3,4}  {2,5}    {2,6}
                       {3,5}    {3,6}
                       {4,5}    {4,6}
                       {1,4,5}  {5,6}
                       {2,3,5}  {1,3,6}
                       {2,4,5}  {1,5,6}
                       {3,4,5}  {2,3,6}
                                {2,4,6}
                                {2,5,6}
                                {3,4,6}
                                {3,5,6}
                                {4,5,6}
                                {1,3,5,6}
                                {1,4,5,6}
                                {2,3,4,6}
                                {2,3,5,6}
                                {2,4,5,6}
                                {3,4,5,6}

The complement with n is counted by A179822, first differences of A326080.
Partial sums are A384350.
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, A326083, A383708, A383710, A384317, A384318, A384319, A384320, A384321.

nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[Subsets[Range[n]],MemberQ[#,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}]

A384395 Number of integer partitions of n with more than one proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 1, 4, 5, 8, 8, 12, 17, 22, 29, 31, 40, 50, 65, 77, 101, 112, 135, 162, 201

Offset: 0

Gus Wiseman, May 30 2025

By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.

			For the partition (8,5,2) we have four choices:
  ((8),(4,1),(2))
  ((7,1),(5),(2))
  ((5,3),(4,1),(2))
  ((4,3,1),(5),(2))
Hence (8,5,2) is counted under a(15).
The a(5) = 1 through a(12) = 12 partitions:
  (5)  (6)    (7)  (8)    (9)    (10)     (11)     (12)
       (3,3)       (4,4)  (5,4)  (5,5)    (6,5)    (6,6)
                   (5,3)  (6,3)  (6,4)    (7,4)    (7,5)
                   (7,1)  (7,2)  (7,3)    (8,3)    (8,4)
                          (8,1)  (8,2)    (9,2)    (9,3)
                                 (9,1)    (10,1)   (10,2)
                                 (4,3,3)  (5,3,3)  (11,1)
                                 (4,4,2)  (5,5,1)  (5,5,2)
                                                   (6,3,3)
                                                   (6,4,2)
                                                   (6,5,1)
                                                   (9,2,1)

For just one choice we have A179009, ranked by A383707.
Twice-partitions of this type are counted by A279790.
For at least one choice we have A383708, odd case A383533.
For no choices we have A383710, odd case A383711.
For at least one proper choice we have A384317, ranked by A384321.
The strict version for at least one proper choice is A384318, ranked by A384322.
The strict version for just one proper choice is A384319, ranked by A384390.
For just one proper choice we have A384323, ranks A384347 = positions of 2 in A383706.
For no proper choices we have A384348, ranked by A384349.
These partitions are ranked by A384393.
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
A357982 counts choices of strict partitions of each prime index, non-strict A299200.
Cf. A098859, A279375, A317142, A381454, A382525, A382912, A382913, A384005.

pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Length[pofprop[#]]>1&]],{n,0,15}]

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A384391 Number of subsets of {1..n} containing n and some element that is a sum of distinct non-elements.

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

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A384395 Number of integer partitions of n with more than one proper way to choose disjoint strict partitions of each part.

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