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

1,2

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=1..49.

Example

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)

Mathematica

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

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.

Sequence in context: A318771 A363726 A239835 * A026929 A206560 A035554

Adjacent sequences: A361857 A361858 A361859 * A361861 A361862 A361863

Keyword

nonn

Author

Gus Wiseman, Apr 02 2023

Status

approved