|
|
A360686
|
|
Number of integer partitions of n whose distinct parts have integer median.
|
|
9
|
|
|
1, 2, 2, 4, 3, 8, 7, 16, 17, 31, 35, 60, 67, 99, 121, 170, 200, 270, 328, 436, 522, 674, 828, 1061, 1292, 1626, 1983, 2507, 3035, 3772, 4582, 5661, 6801, 8358, 10059, 12231, 14627, 17702, 21069, 25423, 30147, 36100, 42725, 50936, 60081, 71388, 84007, 99408
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
EXAMPLE
|
The a(1) = 1 through a(8) = 16 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (311) (33) (331) (44)
(31) (11111) (42) (421) (53)
(1111) (51) (511) (62)
(222) (3211) (71)
(321) (31111) (422)
(3111) (1111111) (431)
(111111) (521)
(2222)
(3221)
(3311)
(4211)
(5111)
(32111)
(311111)
(11111111)
For example, the partition y = (7,4,2,1,1) has distinct parts {1,2,4,7} with median 3, so y is counted under a(15).
|
|
MATHEMATICA
|
Table[Length[Select[IntegerPartitions[n], IntegerQ[Median[Union[#]]]&]], {n, 30}]
|
|
CROSSREFS
|
For multiplicities instead of distinct parts: A360687.
The complement is counted by A360689.
A000975 counts subsets with integer median.
A116608 counts partitions by number of distinct parts.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|