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

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

Author

Gus Wiseman, Jun 07 2025

Keywords

• nonn
• more

Comments

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.

Examples

			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)
		

Crossrefs

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.

Programs

Mathematica

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