A360456 Number of integer partitions of n for which the parts have the same median as the multiplicities.

1, 1, 0, 0, 1, 0, 0, 1, 2, 5, 7, 10, 14, 21, 28, 36, 51, 64, 84, 106, 132, 165, 202, 252, 311, 391, 473, 579, 713, 868, 1069, 1303, 1617, 1954, 2404, 2908, 3556, 4282, 5200, 6207, 7505, 8934, 10700, 12717, 15165, 17863, 21222, 24976, 29443, 34523, 40582, 47415

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) = 10 partitions:
  1   .  .  22   .  .  2221   3311    333      4222      5222
                              32111   3222     33211     33221
                                      32211    42211     52211
                                      42111    43111     53111
                                      321111   52111     62111
                                               421111    322211
                                               3211111   431111
                                                         521111
                                                         4211111
                                                         32111111
		

Crossrefs

For mean instead of median: A360068, ranks A359903.

For distinct parts instead of multiplicities: A360245, ranks A360249.

These partitions have ranks A360454.

For distinct parts instead of parts: A360455, ranks A360453.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by number of parts.

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

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

Cf. A000975, A027193, A114638, A240219, A359894, A359895, A360244, A360248.

Programs

Mathematica

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