A361850 Number of strict integer partitions of n such that the maximum is twice the median.

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 2, 1, 3, 3, 4, 2, 5, 4, 7, 8, 10, 6, 11, 11, 15, 16, 21, 18, 25, 23, 28, 32, 40, 40, 51, 51, 58, 60, 73, 75, 93, 97, 113, 123, 139, 141, 164, 175, 199, 217, 248, 263, 301, 320, 356, 383, 426, 450, 511, 551, 613, 664, 737

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(7) = 1 through a(20) = 4 strict partitions (A..C = 10..12):
  421  .  .  631  632   .  841   842  843   A51    A52    A53   A54   C62
                  5321     6421       7431  7432   8531   8532  C61   9542
                                      7521  64321  8621         9541  9632
                                                   65321        9631  85421
                                                                9721
The partition (7,4,3,1) has maximum 7 and median 7/2, so is counted under a(15).
The partition (8,6,2,1) has maximum 8 and median 4, so is counted under a(17).
		

Crossrefs

For minimum instead of median we have A241035, non-strict A237824.

For length instead of median we have A241087, non-strict A237755.

The non-strict version is A361849, ranks A361856.

The non-strict complement is counted by A361857, ranks A361867.

A000041 counts integer partitions, strict A000009.

A000975 counts subsets with integer median.

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.

A359907 counts strict partitions with integer median

A360005 gives median of prime indices (times two), distinct A360457.

Cf. A027193, A067659, A079309, A111907, A116608, A359897, A359908, A360952, A361851, A361858, A361859, A361860.

Programs

Mathematica

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