A363728 Number of integer partitions of n that are not constant but satisfy (mean) = (median) = (mode), assuming there is a unique mode.

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 4, 0, 3, 3, 7, 0, 12, 0, 18, 12, 9, 0, 50, 12, 14, 33, 54, 0, 115, 0, 92, 75, 31, 99, 323, 0, 45, 162, 443, 0, 507, 0, 467, 732, 88, 0, 1551, 274, 833, 627, 1228, 0, 2035, 1556, 2859, 1152, 221, 0, 9008, 0, 295, 4835, 5358

Offset

1,12

Comments

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}.

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

Links

Table of n, a(n) for n=1..64.

Example

The a(8) = 1 through a(18) = 12 partitions:
  3221  .  32221  .  4332    .  3222221  43332  5443      .  433332
                     5331       3322211  53331  6442         443331
                     322221     4222211  63321  7441         533322
                     422211                     32222221     533331
                                                33222211     543321
                                                42222211     633321
                                                52222111     733311
                                                             322222221
                                                             332222211
                                                             422222211
                                                             432222111
                                                             522222111

Mathematica

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

Crossrefs

Non-constant partitions are counted by A144300, ranks A024619.

This is the non-constant case of A363719, ranks A363727.

These partitions have ranks A363729.

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, A363720, A363721.

Sequence in context: A021252 A348991 A195775 * A362562 A316260 A119708

Adjacent sequences: A363725 A363726 A363727 * A363729 A363730 A363731

Keyword

nonn

Author

Gus Wiseman, Jun 23 2023

Status

approved