A363725 Number of integer partitions of n with a different mean, median, and mode, assuming there is a unique mode.

0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 3, 8, 8, 17, 19, 28, 39, 59, 68, 106, 123, 165, 220, 301, 361, 477, 605, 745, 929, 1245, 1456, 1932, 2328, 2846, 3590, 4292, 5111, 6665, 8040, 9607, 11532, 14410, 16699, 20894, 24287, 28706, 35745, 42845, 49548, 59963, 70985

Offset: 0

Author

Gus Wiseman, Jun 22 2023

Keywords

• nonn

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

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

Examples

			The a(7) = 1 through a(13) = 17 partitions:
  (3211)  (4211)  (3321)  (5311)    (4322)    (4431)    (4432)
                  (4311)  (6211)    (4421)    (5322)    (5422)
                  (5211)  (322111)  (5411)    (6411)    (5521)
                                    (6311)    (7311)    (6322)
                                    (7211)    (8211)    (6511)
                                    (43211)   (53211)   (7411)
                                    (332111)  (432111)  (8311)
                                    (422111)  (522111)  (9211)
                                                        (54211)
                                                        (63211)
                                                        (333211)
                                                        (433111)
                                                        (442111)
                                                        (532111)
                                                        (622111)
                                                        (3322111)
                                                        (32221111)
		

Crossrefs

The length-4 case appears to be A325695.

For equal instead of unequal we have A363719, ranks A363727.

Allowing multiple modes gives A363720, ranks A363730.

A000041 counts partitions, strict A000009.

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

A359893 and A359901 count partitions by median, odd-length A359902.

A362608 counts partitions with a unique mode.

Cf. A237984, A240219, A325347, A326567/A326568, A327472, A359894, A359896, A359900, A363723, A363728.

Programs

Mathematica

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], Length[modes[#]]==1&&Mean[#]!=Median[#]!=First[modes[#]]&]],{n,0,30}]