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

Author

Clark Kimberling, Apr 24 2019

Keywords

• nonn
• easy

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

Examples

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

Links

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

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.

Programs

Mathematica

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