A307683 Number of partitions of n having a non-integer median.

0, 0, 1, 0, 2, 1, 4, 1, 7, 5, 11, 8, 18, 17, 31, 28, 47, 51, 75, 81, 119, 134, 181, 206, 277, 323, 420, 488, 623, 737, 922, 1084, 1352, 1597, 1960, 2313, 2819, 3330, 4029, 4743, 5704, 6722, 8030, 9434, 11234, 13175, 15601, 18262, 21552, 25184, 29612, 34518

Offset

1,5

Comments

This sequence and A325347 partition the partition numbers, A000041.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). - Gus Wiseman, Mar 16 2023

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..180

Example

a(7) counts these 4 partitions: [6,1], [5,2], [4,3], [3,2,1,1].

Mathematica

Table[Count[IntegerPartitions[n], q_ /; !IntegerQ[Median[q]]], {n, 10}]

Crossrefs

The complement is counted by A325347, strict A359907.

For mean instead of median we have A349156, strict A361391.

These partitions have ranks A359912, complement A359908.

The strict case is A360952.

A000041 counts integer partitions, strict A000009.

A008284/A058398/A327482 count partitions by mean.

A359893/A359901/A359902 count partitions by median.

Cf. A000016, A051293, A067538, A082550, A240219, A240850, A316413, A326567/A326568, A327475, A359897, A360005.

Sequence in context: A097360 A360862 A325348 * A248058 A072345 A200583

Adjacent sequences: A307680 A307681 A307682 * A307684 A307685 A307686

Keyword

nonn,easy

Author

Clark Kimberling, Apr 24 2019

Status

approved