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A360689
Number of integer partitions of n whose distinct parts have non-integer median.
1
0, 0, 1, 1, 4, 3, 8, 6, 13, 11, 21, 17, 34, 36, 55, 61, 97, 115, 162, 191, 270, 328, 427, 514, 666, 810, 1027, 1211, 1530, 1832, 2260, 2688, 3342, 3952, 4824, 5746, 7010, 8313, 10116, 11915, 14436, 17074, 20536, 24239, 29053, 34170, 40747, 47865, 56830, 66621
OFFSET
1,5
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).
EXAMPLE
The a(1) = 0 through a(9) = 13 partitions:
. . (21) (211) (32) (411) (43) (332) (54)
(41) (2211) (52) (611) (63)
(221) (21111) (61) (22211) (72)
(2111) (322) (41111) (81)
(2221) (221111) (441)
(4111) (2111111) (522)
(22111) (3222)
(211111) (6111)
(22221)
(222111)
(411111)
(2211111)
(21111111)
For example, the partition y = (5,3,3,2,1,1) has distinct parts {1,2,3,5}, with median 5/2, so y is counted under a(15).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Union[#]]]&]], {n, 30}]
CROSSREFS
For not just distinct parts: A307683, complement A325347, ranks A359912.
These partitions have ranks A360551.
The complement is counted by A360686, strict A359907, ranks A360550.
For multiplicities instead of distinct parts we have A360690, ranks A360554.
A000041 counts integer partitions, strict A000009.
A116608 counts partitions by number of distinct parts.
A359893 and A359901 count partitions by median, odd-length A359902.
A360457 gives median of distinct prime indices (times 2).
Sequence in context: A330686 A117956 A241638 * A110662 A265289 A302258
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 22 2023
STATUS
approved