A359907 Number 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

Offset: 0

Author

Gus Wiseman, Jan 21 2023

Keywords

• nonn

Comments

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

Examples

			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
		

Crossrefs

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

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

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

A000041 counts partitions, strict A000009.

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

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

A326567/A326568 gives the mean of prime indices.

A359893, A359901, A359902 count partitions by median.

Cf. A000016, A082550, A240219, A240850, A326669, A327475, A328966, A359897.

Programs

Mathematica

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