A363719 - OEIS

%I #19 Jun 22 2023 18:45:00

%S 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,

%T 123,2,98,79,35,103,332,2,49,166,451,2,515,2,473,738,92,2,1561,277,

%U 839,631,1234,2,2043,1560,2867,1156,225,2,9020

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

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

%C 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}.

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

%C Includes all constant partitions.

%e The a(n) partitions for n = 1, 2, 4, 6, 8, 12, 14, 16 (A..G = 10..16):

%e   1  2   4     6       8         C             E               G

%e      11  22    33      44        66            77              88

%e          1111  222     2222      444           2222222         4444

%e                111111  3221      3333          3222221         5443

%e                        11111111  4332          3322211         6442

%e                                  5331          4222211         7441

%e                                  222222        11111111111111  22222222

%e                                  322221                        32222221

%e                                  422211                        33222211

%e                                  111111111111                  42222211

%e                                                                52222111

%e                                                                1^16

%t modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];

%t Table[Length[Select[IntegerPartitions[n], {Mean[#]}=={Median[#]}==modes[#]&]],{n,30}]

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

%Y The odd-length case is A363721.

%Y These partitions have ranks A363727, nonprime A363722.

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

%Y The version for factorizations is A363741, see A359909, A359910.

%Y Just two statistics:

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

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

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

%Y - (median) = (mode) gives A363740.

%Y A000041 counts partitions, strict A000009.

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

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

%Y A362608 counts partitions with a unique mode.

%Y Cf. A027193, A237984, A325347, A326567/A326568, A327472, A363726, A363742.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 19 2023