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

Author

Gus Wiseman, Feb 20 2023

Keywords

• nonn

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.

Examples

			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).
		

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.

Programs

Mathematica

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