A363721 - OEIS

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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

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OFFSET

1,3

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}.

LINKS

Table of n, a(n) for n=1..60.

EXAMPLE

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

MATHEMATICA

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

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

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.

Sequence in context: A355748 A295894 A068307 * A158946 A303428 A223853

Adjacent sequences: A363718 A363719 A363720 * A363722 A363723 A363724

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 21 2023

STATUS

approved