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A362558
Number of integer partitions of n without a nonempty initial consecutive subsequence summing to n/2.
5
1, 1, 1, 3, 2, 7, 6, 15, 11, 30, 27, 56, 44, 101, 93, 176, 149, 297, 271, 490, 432, 792, 744, 1255, 1109, 1958, 1849, 3010, 2764, 4565, 4287, 6842, 6328, 10143, 9673, 14883, 13853, 21637, 20717, 31185, 29343, 44583, 42609, 63261, 60100, 89134, 85893, 124754
OFFSET
0,4
COMMENTS
Also the number of n-multisets of positive integers that (1) have integer median, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.
EXAMPLE
The a(1) = 1 through a(7) = 15 partitions:
(1) (2) (3) (4) (5) (6) (7)
(21) (31) (32) (42) (43)
(111) (41) (51) (52)
(221) (222) (61)
(311) (411) (322)
(2111) (2211) (331)
(11111) (421)
(511)
(2221)
(3211)
(4111)
(22111)
(31111)
(211111)
(1111111)
The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(8).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !MemberQ[Accumulate[#], n/2]&]], {n, 0, 15}]
CROSSREFS
The odd bisection is A058695.
The version for compositions is A213173.
The complement is counted by A322439 aerated.
The even bisection is A362051.
For mean instead of median we have A362559.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
A359893/A359901/A359902 count partitions by median.
Sequence in context: A332811 A286556 A243354 * A321184 A305123 A364787
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 24 2023
STATUS
approved