A363723 Number of integer partitions of n having a unique mode equal to the mean, i.e., partitions whose mean appears more times than each of the other parts.

0, 1, 2, 2, 3, 2, 4, 2, 5, 3, 5, 2, 10, 2, 7, 7, 12, 2, 18, 2, 24, 16, 13, 2, 60, 15, 18, 37, 60, 2, 129, 2, 104, 80, 35, 104, 352, 2, 49, 168, 501, 2, 556, 2, 489, 763, 92, 2, 1799, 292, 985, 649, 1296, 2, 2233, 1681, 3379, 1204, 225, 2, 10661

Offset: 0

Author

Gus Wiseman, Jun 24 2023

Keywords

• nonn

Comments

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

Examples

			The a(n) partitions for n = 6, 8, 12, 14, 16 (A..G = 10..16):
  (6)       (8)         (C)             (E)               (G)
  (33)      (44)        (66)            (77)              (88)
  (222)     (2222)      (444)           (2222222)         (4444)
  (111111)  (3221)      (3333)          (3222221)         (5443)
            (11111111)  (4332)          (3322211)         (6442)
                        (5331)          (4222211)         (7441)
                        (222222)        (11111111111111)  (22222222)
                        (322221)                          (32222221)
                        (422211)                          (33222211)
                        (111111111111)                    (42222211)
                                                          (52222111)
                                                          (1111111111111111)
		

Crossrefs

Partitions containing their mean are counted by A237984, ranks A327473.

For median instead of mode we have A240219, ranks A359889.

Partitions missing their mean are counted by A327472, ranks A327476.

The case of non-constant partitions is A362562.

Including median also gives A363719, ranks A363727.

Allowing multiple modes gives A363724.

Requiring multiple modes gives A363731.

For median instead of mean we have A363740.

A000041 counts partitions, strict A000009.

A008284 counts partitions by length (or decreasing mean), strict A008289.

A359893 and A359901 count partitions by median.

A362608 counts partitions with a unique mode.

Cf. A325347, A326567/A326568, A363720, A363725, A363730.

Programs

Mathematica

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],{Mean[#]}==modes[#]&]],{n,30}]