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proposed

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

Table[Length[Select[IntegerPartitions[n], IntegerQ[Mean[#]]&&IntegerQ[Median[#]]&]], {n, 1, 30}]

A008284/A058398/A327482 count counts partitions by mean, see also A008284, A327482.

A237984 counts partitions containing their mean, strict A240850, ranked by A327473.

A359893/A359901/A359902 count partitions by median, ranked by A360005.

A360005(n)/2 gives median of prime indices.

Cf. A000016, A082550, A237984, A240219, `A316313, A326669, A327475, A349156, `A359889, A359894, A359897, A359905.

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

allocated for Gus WisemanNumber of integer partitions of n with integer mean and integer median.

1, 2, 2, 4, 2, 8, 2, 10, 9, 14, 2, 39, 2, 24, 51, 49, 2, 109, 2, 170, 144, 69, 2, 455, 194, 116, 381, 668, 2, 1378, 2, 985, 956, 316, 2043, 4328, 2, 511, 2293, 6656, 2, 8634, 2, 8062, 14671, 1280, 2, 26228, 8035, 15991, 11614, 25055, 2, 47201, 39810, 65092

1,2

The a(1) = 1 through a(9) = 9 partitions:

1 2 3 4 5 6 7 8 9

11 111 22 11111 33 1111111 44 333

31 42 53 432

1111 51 62 441

222 71 522

321 2222 531

411 3221 621

111111 3311 711

5111 111111111

11111111

Table[Length[Select[IntegerPartitions[n], IntegerQ[Mean[#]]&&IntegerQ[Median[#]]&]], {n, 1, 30}]

For just integer mean we have A067538, strict A102627, ranked by A316413.

For just integer median we have A325347, strict A359907, ranked by A359908.

These partitions are ranked by A360009.

A000041 counts partitions, strict A000009.

A008284/A058398/A327482 count partitions by mean.

A051293 counts subsets with integer mean, median A000975.

A237984 counts partitions containing their mean, strict A240850, ranked by A327473.

A326567/A326568 gives mean of prime indices.

A326622 counts factorizations with integer mean, strict A328966.

A359893/A359901/A359902 count partitions by median, ranked by A360005.

Cf. A000016, A082550, A240219, `A316313, A326669, A327475, A349156, `A359889, A359894, A359897, A359905.

allocated

nonn

Gus Wiseman, Jan 21 2023

approved

editing

allocated for Gus Wiseman

allocated

approved