A360455 Number of integer partitions of n for which the distinct parts have the same median as the multiplicities.

1, 1, 0, 0, 2, 1, 1, 0, 2, 2, 5, 8, 10, 14, 20, 19, 26, 31, 35, 41, 55, 65, 85, 102, 118, 151, 181, 201, 236, 281, 313, 365, 424, 495, 593, 688, 825, 978, 1181, 1374, 1650, 1948, 2323, 2682, 3175, 3680, 4314, 4930, 5718, 6546, 7532, 8557, 9777, 11067, 12622

Offset: 0

Author

Gus Wiseman, Feb 10 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(1) = 1 through a(11) = 8 partitions:
  1   .  .  22    221   3111   .  3311    333     3331     32222
            211                   41111   32211   33211    33221
                                                  42211    44111
                                                  322111   52211
                                                  511111   322211
                                                           332111
                                                           422111
                                                           3221111
		

Crossrefs

For mean instead of median: A114638, ranks A324570.

For parts instead of multiplicities: A360245, ranks A360249.

These partitions have ranks A360453.

For parts instead of distinct parts: A360456, ranks A360454.

A000041 counts integer partitions, strict A000009.

A116608 counts partitions by number of distinct parts.

A325347 counts partitions w/ integer median, strict A359907, ranks A359908.

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

Cf. A000975, A027193, A240219, A326619/A326620, A359894, A359895, A360068, A360243, A360244, A360248.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], Median[Length/@Split[#]]==Median[Union[#]]&]],{n,0,30}]