A363731 Number of integer partitions of n whose mean is a mode but not the only mode.

0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 5, 0, 1, 8, 5, 0, 12, 0, 19, 14, 2, 0, 52, 21, 3, 23, 59, 0, 122, 0, 97, 46, 6, 167, 303, 0, 8, 82, 559, 0, 543, 0, 355, 745, 15, 0, 1685, 510, 1083, 251, 840, 0, 2325, 1832, 3692, 426, 34, 0, 9599

Offset

0,10

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(n) partitions for n = 6, 9, 12, 15, 18:
  (3,2,1)  (4,3,2)  (5,4,3)        (6,5,4)      (7,6,5)
           (5,3,1)  (6,4,2)        (7,5,3)      (8,6,4)
                    (7,4,1)        (8,5,2)      (9,6,3)
                    (6,3,2,1)      (9,5,1)      (10,6,2)
                    (3,3,2,2,1,1)  (4,4,3,3,1)  (11,6,1)
                                   (5,3,3,2,2)  (4,4,3,3,2,2)
                                   (5,4,3,2,1)  (5,5,3,3,1,1)
                                   (7,3,3,1,1)  (6,4,3,3,1,1)
                                                (7,3,3,2,2,1)
                                                (8,3,3,2,1,1)
                                                (3,3,3,2,2,2,1,1,1)
                                                (6,2,2,2,2,1,1,1,1)

Mathematica

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

Crossrefs

For a unique mode we have A363723, non-constant A362562.

For any number of modes we have A363724.

A000041 counts partitions, strict A000009.

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

A237984 counts partitions containing their mean, ranks A327473.

A327472 counts partitions not containing their mean, ranks A327476.

A362608 counts partitions with a unique mode, ranks A356862.

A363719 counts partitions with all three averages equal, ranks A363727.

Cf. A240219, A326567/A326568, A359893, A363720, A363725, A363740.

Sequence in context: A254882 A086095 A322119 * A112334 A113469 A060137

Adjacent sequences: A363728 A363729 A363730 * A363732 A363733 A363734

Keyword

nonn

Author

Gus Wiseman, Jun 24 2023

Status

approved