A361860 Number of integer partitions of n whose median part is the smallest.

1, 2, 2, 4, 4, 7, 8, 12, 15, 21, 25, 36, 44, 58, 72, 95, 117, 150, 185, 235, 289, 362, 441, 550, 670, 824, 1000, 1223, 1476, 1795, 2159, 2609, 3126, 3758, 4485, 5369, 6388, 7609, 9021, 10709, 12654, 14966, 17632, 20782, 24414, 28684, 33601, 39364, 45996

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(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (211)   (2111)   (222)     (511)      (422)
                    (1111)  (11111)  (411)     (4111)     (611)
                                     (3111)    (22111)    (2222)
                                     (21111)   (31111)    (5111)
                                     (111111)  (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
		

Crossrefs

For mean instead of median we have A000005.

For length instead of median we have A006141.

For maximum instead of median we have A053263.

For half-median we have A361861.

A000041 counts integer partitions, strict A000009.

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.

A360005 gives twice median of prime indices, distinct A360457.

Cf. A027193, A053263, A067659, A111907, A116608, A118096, A237753, A240219, A359907, A361848, A361849.

Programs

Mathematica

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