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

0,5

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).

Links

Table of n, a(n) for n=0..54.

Example

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

Mathematica

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

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.

Sequence in context: A366388 A114638 A123340 * A267486 A285229 A227425

Adjacent sequences: A360452 A360453 A360454 * A360456 A360457 A360458

Keyword

nonn

Author

Gus Wiseman, Feb 10 2023

Status

approved