A363719 Number of integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

1, 2, 2, 3, 2, 4, 2, 5, 3, 5, 2, 10, 2, 7, 7, 12, 2, 18, 2, 24, 16, 13, 2, 58, 15, 18, 37, 60, 2, 123, 2, 98, 79, 35, 103, 332, 2, 49, 166, 451, 2, 515, 2, 473, 738, 92, 2, 1561, 277, 839, 631, 1234, 2, 2043, 1560, 2867, 1156, 225, 2, 9020

Offset: 1

Author

Gus Wiseman, Jun 19 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}.

Without loss of generality, we may assume there is a unique middle-part (A238478).

Includes all constant partitions.

Examples

			The a(n) partitions for n = 1, 2, 4, 6, 8, 12, 14, 16 (A..G = 10..16):
  1  2   4     6       8         C             E               G
     11  22    33      44        66            77              88
         1111  222     2222      444           2222222         4444
               111111  3221      3333          3222221         5443
                       11111111  4332          3322211         6442
                                 5331          4222211         7441
                                 222222        11111111111111  22222222
                                 322221                        32222221
                                 422211                        33222211
                                 111111111111                  42222211
                                                               52222111
                                                               1^16
		

Crossrefs

For unequal instead of equal: A363720, ranks A363730, unique mode A363725.

The odd-length case is A363721.

These partitions have ranks A363727, nonprime A363722.

The case of non-constant partitions is A363728, ranks A363729.

The version for factorizations is 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, A363742.

Programs

Mathematica

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