A360688 Number of integer partitions of n with integer median of 0-appended first differences.

1, 1, 3, 4, 5, 7, 12, 18, 25, 32, 46, 62, 79, 109, 142, 189, 240, 322, 405, 522, 671, 853, 1053, 1345, 1653, 2081, 2551, 3174, 3878, 4826, 5851, 7219, 8747, 10712, 12936, 15719, 18876, 22872, 27365, 32926, 39253, 47070, 55857, 66676, 79029, 93864, 110832

Offset: 1

Author

Gus Wiseman, Feb 20 2023

Keywords

• nonn

Comments

Includes all partitions of odd length (A027193).

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The a(1) = 1 through a(8) = 18 partitions:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)        (8)
            (21)   (22)    (41)     (42)      (43)       (44)
            (111)  (211)   (221)    (222)     (61)       (62)
                   (1111)  (311)    (321)     (322)      (332)
                           (11111)  (411)     (331)      (422)
                                    (21111)   (421)      (431)
                                    (111111)  (511)      (521)
                                              (3211)     (611)
                                              (22111)    (2222)
                                              (31111)    (3221)
                                              (211111)   (4211)
                                              (1111111)  (22211)
                                                         (32111)
                                                         (41111)
                                                         (221111)
                                                         (311111)
                                                         (2111111)
                                                         (11111111)
For example, the partition y = (3,2,2,1) has 0-appended parts (3,2,2,1,0), with differences (1,0,1,1), and the multiset {0,1,1,1} has median 1, so y is counted under a(8).
		

Crossrefs

The case of median 0 is A360254, ranks A360558.

These partitions have ranks A360556, complement A360557.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by number of parts.

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

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

Cf. A000975, A027193, A237363, A240219, A360455, A360555.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],IntegerQ[Median[Differences[Prepend[Reverse[#],0]]]]&]],{n,30}]