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

0,13

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}.

Links

Table of n, a(n) for n=0..60.

Example

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)

Mathematica

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

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.

Sequence in context: A348991 A195775 A363728 * A316260 A119708 A106142

Adjacent sequences: A362559 A362560 A362561 * A362563 A362564 A362565

Keyword

nonn

Author

Gus Wiseman, Jun 27 2023

Status

approved