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

For mean instead of median we have : A102627, non-strict A067538 (ranked by A316413).

The median statistic is ranked by A360005(n)/2.

A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.

A359893/A359901/A359902 count A058398 counts partitions by median, ranked by A360005mean, see also A008284, A327482.

A326567/A326568 gives the mean of prime indices.

A359893, A359901, A359902 count partitions by median.

Cf. A000016, A082550, A240219, A240850, `A316313, `A326669, A327475, A328966, `A349156, A359897.

The median of a multiset is the middle part in the odd-length case, and the average of the two middle parts in the even-length case.

allocated for Gus WisemanNumber of strict integer partitions of n with integer median.

0, 1, 1, 1, 2, 1, 4, 2, 6, 4, 9, 6, 14, 10, 18, 16, 27, 23, 36, 34, 51, 49, 67, 68, 94, 95, 122, 129, 166, 174, 217, 233, 287, 308, 371, 405, 487, 528, 622, 683, 805, 880, 1024, 1127, 1305, 1435, 1648, 1818, 2086, 2295, 2611, 2882, 3273, 3606, 4076, 4496, 5069

0,5

The a(1) = 1 through a(14) = 18 partitions (A..E = 10..14):

1 2 3 4 5 6 7 8 9 A B C D E

31 42 421 53 432 64 542 75 643 86

51 62 531 73 632 84 652 95

321 71 621 82 641 93 742 A4

431 91 731 A2 751 B3

521 532 821 B1 832 C2

541 543 841 D1

631 642 931 653

721 651 A21 743

732 6421 752

741 761

831 842

921 851

5421 932

941

A31

B21

7421

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&IntegerQ[Median[#]]&]], {n, 0, 30}]

For mean instead of median we have A102627, non-strict A067538 (ranked by A316413).

This is the strict case of A325347, ranked by A359908.

A000041 counts partitions, strict A000009.

A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.

A051293 counts subsets with integer mean, median A000975, cf. A005578.

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

Cf. A000016, A082550, A240219, A240850, `A316313, `A326669, A327475, A328966, `A349156, A359897.

allocated

nonn

Gus Wiseman, Jan 21 2023

approved

editing

allocated for Gus Wiseman

allocated

approved