A361849 Number of integer partitions of n such that the maximum is twice the median.

0, 0, 0, 1, 1, 1, 4, 3, 4, 7, 9, 9, 15, 16, 20, 26, 34, 37, 50, 55, 68, 86, 103, 117, 145, 168, 201, 236, 282, 324, 391, 449, 525, 612, 712, 818, 962, 1106, 1278, 1470, 1698, 1939, 2238, 2550, 2924, 3343, 3824, 4341, 4963, 5627, 6399, 7256, 8231, 9300

Offset: 1

Author

Gus Wiseman, Apr 02 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(4) = 1 through a(11) = 9 partitions:
  211  2111  21111  421     422      4221      631        632
                    3211    221111   4311      4222       5321
                    22111   2111111  2211111   42211      5411
                    211111           21111111  322111     42221
                                               2221111    43211
                                               22111111   332111
                                               211111111  22211111
                                                          221111111
                                                          2111111111
For example, the partition (3,2,1,1) has maximum 3 and median 3/2, so is counted under a(7).
		

Crossrefs

For minimum instead of median we have A118096.

For length instead of median we have A237753.

This is the equal case of A361848.

For mean instead of median we have A361853.

These partitions have ranks A361856.

For "greater" instead of "equal" we have A361857, allowing equality A361859.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, A058398 by mean.

A325347 counts partitions with integer median, complement A307683.

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

A360005 gives twice median of prime indices, distinct A360457.

A361860 counts partitions with minimum equal to median.

Cf. A013580, A027193, A053263, A061395, A067659, A111907, A116608, A237755, A240219, A361851.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Max@@#==2*Median[#]&]],{n,30}]