A363728 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

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

(list; graph; refs; listen; history; text; internal format)

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