A361800 Number of integer partitions of n with the same length as median.

1, 0, 0, 2, 0, 0, 1, 2, 3, 3, 3, 3, 4, 6, 9, 13, 14, 15, 18, 21, 27, 32, 40, 46, 55, 62, 72, 82, 95, 111, 131, 157, 186, 225, 264, 316, 366, 430, 495, 578, 663, 768, 880, 1011, 1151, 1316, 1489, 1690, 1910, 2158, 2432, 2751, 3100, 3505, 3964, 4486, 5079, 5764

Offset: 1

Author

Gus Wiseman, Apr 07 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(15) = 9 partitions (A=10, B=11):
  1  .  .  22  .  .  331  332  333  433  533  633  733   833   933
           31             431  432  532  632  732  832   932   A32
                               531  631  731  831  931   A31   B31
                                                   4441  4442  4443
                                                         5441  5442
                                                         5531  5532
                                                               6441
                                                               6531
                                                               6621
		

Crossrefs

For minimum instead of median we have A006141, for twice minimum A237757.

For maximum instead of median we have A047993, for twice length A237753.

For maximum instead of length we have A053263, for twice median A361849.

For mean instead of median we have A206240 (zeros removed).

For minimum instead of length we have A361860.

For twice median we have A362049, ranks A362050.

A000041 counts integer partitions, strict A000009.

A000975 counts subsets with integer median.

A325347 counts partitions with integer median, complement A307683.

A359893 and A359901 count partitions by median.

A360005 gives twice median of prime indices.

Cf. A008284, A013580, A027193, A079309, A240219, A362048.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Length[#]==Median[#]&]],{n,30}]