A360254 Number of integer partitions of n with more adjacent equal parts than distinct parts.

0, 0, 0, 1, 1, 1, 3, 4, 7, 10, 12, 18, 28, 36, 52, 68, 92, 119, 161, 204, 269, 355, 452, 571, 738, 921, 1167, 1457, 1829, 2270, 2834, 3483, 4314, 5300, 6502, 7932, 9665, 11735, 14263, 17227, 20807, 25042, 30137, 36099, 43264, 51646, 61608, 73291, 87146, 103296

Offset

0,7

Comments

None of these partitions is strict.

Also the number of integer partitions of n which, after appending 0, have first differences of median 0.

Links

Table of n, a(n) for n=0..49.

Example

The a(3) = 1 through a(9) = 10 partitions:
  (111)  (1111)  (11111)  (222)     (22111)    (2222)      (333)
                          (21111)   (31111)    (22211)     (22221)
                          (111111)  (211111)   (41111)     (33111)
                                    (1111111)  (221111)    (51111)
                                               (311111)    (222111)
                                               (2111111)   (411111)
                                               (11111111)  (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)
For example, the partition y = (4,4,3,1,1,1,1) has 0-appended differences (0,1,2,0,0,0,0), with median 0, so y is counted under a(15).

Mathematica

Table[Length[Select[IntegerPartitions[n], Length[#]>2*Length[Union[#]]&]], {n, 0, 30}]

Crossrefs

The non-prepended version is A237363.

These partitions have ranks A360558.

For any integer median (not just 0) we have A360688.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by number of parts.

A116608 counts partitions by number of distinct parts.

A325347 counts partitions w/ integer median, strict A359907, ranks A359908.

A359893 and A359901 count partitions by median, odd-length A359902.

Cf. A000975, A027193, A067538, A102627, A240219, A359894, A360071, A360244, A360555, A360556.

Sequence in context: A282573 A177959 A371634 * A108855 A026488 A108579

Adjacent sequences: A360251 A360252 A360253 * A360255 A360256 A360257

Keyword

nonn

Author

Gus Wiseman, Feb 20 2023

Status

approved