A360691 Number of integer partitions of n with non-integer median of 0-prepended first differences.

0, 1, 0, 1, 2, 4, 3, 4, 5, 10, 10, 15, 22, 26, 34, 42, 57, 63, 85, 105, 121, 149, 202, 230, 305, 355, 459, 544, 687, 778, 991, 1130, 1396, 1598, 1947, 2258, 2761, 3143, 3820, 4412, 5330, 6104, 7404, 8499, 10105, 11694, 13922, 15917, 18904, 21646, 25462, 29213

Offset: 1

Author

Gus Wiseman, Feb 22 2023

Keywords

• nonn

Comments

All of these partitions have even length.

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(10) = 10 partitions:
  .  (11)  .  (31)  (32)    (33)    (52)    (53)    (54)      (55)
                    (2111)  (51)    (2221)  (71)    (72)      (73)
                            (2211)  (4111)  (3311)  (3222)    (91)
                            (3111)          (5111)  (6111)    (3322)
                                                    (321111)  (3331)
                                                              (4411)
                                                              (5311)
                                                              (7111)
                                                              (322111)
                                                              (421111)
		

Crossrefs

For median 0 we have A360254, ranks A360558.

These partitions have ranks A360557, complement A360556.

The complement is counted by A360688.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by number of parts.

A325347 counts partitions with integer median, complement A307683.

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

Cf. A000975, A027193, A237363, A240219, A359907, A359908, A360455, A360555.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Differences[Prepend[Reverse[#],0]]]]&]],{n,30}]