A389811 Number of integer partitions of n whose parts are not in arithmetic progression.

0, 0, 0, 0, 1, 3, 4, 10, 15, 21, 33, 49, 63, 93, 124, 160, 218, 287, 365, 479, 610, 771, 986, 1242, 1548, 1941, 2418, 2984, 3696, 4549, 5569, 6825, 8326, 10112, 12287, 14858, 17936, 21617, 25990, 31149, 37305, 44561, 53128, 63238, 75144, 89086, 105528, 124729

Offset

0,6

Comments

These are partitions whose first differences are not all equal.

Links

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

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

The a(0) = 0 through a(8) = 15 partitions:
  .  .  .  .  (211)  (221)   (411)    (322)     (332)
                     (311)   (2211)   (331)     (422)
                     (2111)  (3111)   (421)     (431)
                             (21111)  (511)     (521)
                                      (2221)    (611)
                                      (3211)    (3221)
                                      (4111)    (3311)
                                      (22111)   (4211)
                                      (31111)   (5111)
                                      (211111)  (22211)
                                                (32111)
                                                (41111)
                                                (221111)
                                                (311111)
                                                (2111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], !SameQ@@Differences[#]&]], {n, 0, 15}]

Crossrefs

The complement is A049988, ranks A325328.

The complement for compositions is A175342, ranks A389731, subsets A051336.

For compositions instead of partitions we have A389741, ranks A389735.

These partitions are ranked by A389812.

For distinct instead of equal differences we have A389919.

The 0-appended version is A389920, complement A007862.

A000041 counts integer partitions, strict A000009.

A000837 counts aperiodic partitions.

A011782 counts compositions.

Cf. A070211, A325325, A325545, A325546, A325557, A332834, A389742, A389743.

Sequence in context: A319719 A347568 A048155 * A242342 A204292 A218277

Adjacent sequences: A389807 A389809 A389810 * A389812 A389813 A389814

Keyword

nonn

Author

Gus Wiseman, Oct 19 2025

Status

approved