A363720 Number of integer partitions of n with different mean, median, and mode.

0, 0, 0, 0, 0, 0, 0, 2, 3, 5, 7, 16, 17, 34, 38, 50, 79, 115, 123, 198, 220, 291, 399, 536, 605, 815, 1036, 1241, 1520, 2059, 2315, 3132, 3708, 4491, 5668, 6587, 7788, 10259, 12299, 14515, 17153, 21558, 24623, 30876, 35540, 41476, 52023, 61931, 70811, 85545

Offset: 0

Author

Gus Wiseman, Jun 21 2023

Keywords

• nonn

Comments

If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.

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) = 2 through a(11) = 16 partitions:
  (421)   (431)   (621)    (532)     (542)
  (3211)  (521)   (3321)   (541)     (632)
          (4211)  (4311)   (631)     (641)
                  (5211)   (721)     (731)
                  (32211)  (5311)    (821)
                           (6211)    (4322)
                           (322111)  (4421)
                                     (5321)
                                     (5411)
                                     (6311)
                                     (7211)
                                     (33221)
                                     (43211)
                                     (52211)
                                     (332111)
                                     (422111)
		

Crossrefs

For equal instead of unequal: A363719, ranks A363727, odd-length A363721.

The case of a unique mode is A363725.

These partitions have ranks A363730.

For factorizations we have A363742, for equal A363741, see A359909, A359910.

Just two statistics:

- (mean) = (median) gives A240219, also A359889, A359895, A359897, A359899.

- (mean) != (median) gives A359894, also A359890, A359896, A359898, A359900.

- (mean) = (mode) gives A363723, see A363724, A363731.

- (median) = (mode) gives A363740.

A000041 counts partitions, strict A000009.

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

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

A362608 counts partitions with a unique mode.

Cf. A027193, A237984, A325347, A326567/A326568, A327472, A363726, A363728.

Programs

Mathematica

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