A362562 Number of non-constant integer partitions of n having a unique mode equal to the mean.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 4, 0, 3, 3, 7, 0, 12, 0, 18, 12, 9, 0, 52, 12, 14, 33, 54, 0, 121, 0, 98, 76, 31, 100, 343, 0, 45, 164, 493, 0, 548, 0, 483, 757, 88, 0, 1789, 289, 979, 645, 1290, 0, 2225, 1677, 3371, 1200, 221, 0, 10649

Offset: 0

Author

Gus Wiseman, Jun 27 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(8) = 1 through a(16) = 7 partitions:
  (3221)  .  (32221)  .  (4332)    .  (3222221)  (43332)  (5443)
                         (5331)       (3322211)  (53331)  (6442)
                         (322221)     (4222211)  (63321)  (7441)
                         (422211)                         (32222221)
                                                          (33222211)
                                                          (42222211)
                                                          (52222111)
		

Crossrefs

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

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

Allowing constant partitions gives A363723.

Including median also gives A363728, ranks A363729.

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. A240219, A325347, A363719, A363720, A363724, A363725, A363731, A363740.

Programs

Mathematica

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&{Mean[#]}==modes[#]&]],{n,0,30}]