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

1,5

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).

Links

Table of n, a(n) for n=1..52.

Example

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)

Mathematica

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

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.

Sequence in context: A083172 A287797 A369430 * A302707 A355399 A224714

Adjacent sequences: A360688 A360689 A360690 * A360692 A360693 A360694

Keyword

nonn

Author

Gus Wiseman, Feb 22 2023

Status

approved