A363721 Number of odd-length integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

1, 1, 2, 1, 2, 2, 2, 1, 3, 3, 2, 2, 2, 5, 7, 1, 2, 8, 2, 9, 16, 11, 2, 2, 15, 16, 37, 33, 2, 44, 2, 1, 79, 33, 103, 127, 2, 47, 166, 39, 2, 214, 2, 384, 738, 90, 2, 2, 277, 185, 631, 1077, 2, 1065, 1560, 477, 1156, 223, 2, 2863

Offset: 1

Author

Gus Wiseman, Jun 21 2023

Keywords

• nonn

Comments

The median of an odd-length partition is the middle part.

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(n) partitions for n = {1, 3, 9, 14, 15, 18, 20, 22} (A..M = 10..22):
  1  3    9          E        F                I          K      M
     111  333        2222222  555              666        44444  22222222222
          111111111  3222221  33333            222222222  54443  32222222221
                     3322211  43332            322222221  64442  33222222211
                     4222211  53331            332222211  65441  33322222111
                              63321            422222211  74432  42222222211
                              111111111111111  432222111  74441  43222222111
                                               522222111  84431  44222221111
                                                          94421  52222222111
                                                                 53222221111
                                                                 62222221111
		

Crossrefs

All odd-length partitions are counted by A027193.

For just (mean) = (median) we have A359895, also A240219, A359899, A359910.

For just (mean) != (median) we have A359896, also A359894, A359900.

Allowing any length gives A363719, ranks A363727, non-constant A363728.

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.

A363726 counts odd-length partitions with a unique mode.

Cf. A237984, A325347, A326567/A326568, A327472, A363720, A363740, A363741.

Programs

Mathematica

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