A360689 Number of integer partitions of n whose distinct parts have non-integer median.

0, 0, 1, 1, 4, 3, 8, 6, 13, 11, 21, 17, 34, 36, 55, 61, 97, 115, 162, 191, 270, 328, 427, 514, 666, 810, 1027, 1211, 1530, 1832, 2260, 2688, 3342, 3952, 4824, 5746, 7010, 8313, 10116, 11915, 14436, 17074, 20536, 24239, 29053, 34170, 40747, 47865, 56830, 66621

Offset: 1

Author

Gus Wiseman, Feb 22 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) = 0 through a(9) = 13 partitions:
  .  .  (21)  (211)  (32)    (411)    (43)      (332)      (54)
                     (41)    (2211)   (52)      (611)      (63)
                     (221)   (21111)  (61)      (22211)    (72)
                     (2111)           (322)     (41111)    (81)
                                      (2221)    (221111)   (441)
                                      (4111)    (2111111)  (522)
                                      (22111)              (3222)
                                      (211111)             (6111)
                                                           (22221)
                                                           (222111)
                                                           (411111)
                                                           (2211111)
                                                           (21111111)
For example, the partition y = (5,3,3,2,1,1) has distinct parts {1,2,3,5}, with median 5/2, so y is counted under a(15).
		

Crossrefs

For not just distinct parts: A307683, complement A325347, ranks A359912.

These partitions have ranks A360551.

The complement is counted by A360686, strict A359907, ranks A360550.

For multiplicities instead of distinct parts we have A360690, ranks A360554.

A000041 counts integer partitions, strict A000009.

A116608 counts partitions by number of distinct parts.

A359893 and A359901 count partitions by median, odd-length A359902.

A360457 gives median of distinct prime indices (times 2).

Cf. A000975, A027193, A090794, A240219, A349156, A360005, A360071, A360241, A360244, A360245.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],!IntegerQ[Median[Union[#]]]&]],{n,30}]