A362558 Number of integer partitions of n without a nonempty initial consecutive subsequence summing to n/2.

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.

Links

Table of n, a(n) for n=0..47.

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.

Cf. A058398, A108917, A169942, A325676, A353864, A360254, A360672, A360675, A360686, A360687, A362560.

Sequence in context: A332811 A286556 A243354 * A321184 A305123 A364787

Adjacent sequences: A362555 A362556 A362557 * A362559 A362560 A362561

Keyword

nonn

Author

Gus Wiseman, Apr 24 2023

Status

approved