A389919 Number of integer partitions of n whose first differences are not all distinct.

0, 0, 0, 1, 1, 2, 6, 7, 11, 18, 26, 34, 56, 71, 101, 134, 182, 233, 318, 403, 532, 675, 870, 1095, 1406, 1751, 2206, 2736, 3417, 4205, 5209, 6379, 7843, 9541, 11654, 14121, 17143, 20677, 24973, 29965, 36027, 43078, 51531, 61402, 73175, 86793, 103067, 121927

Offset

0,6

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(3) = 1 through a(9) = 18 partitions:
  (111)  (1111)  (2111)   (222)     (2221)     (2222)      (333)
                 (11111)  (321)     (3211)     (3221)      (432)
                          (2211)    (4111)     (3311)      (531)
                          (3111)    (22111)    (5111)      (3222)
                          (21111)   (31111)    (22211)     (3321)
                          (111111)  (211111)   (32111)     (6111)
                                    (1111111)  (41111)     (22221)
                                               (221111)    (32211)
                                               (311111)    (33111)
                                               (2111111)   (42111)
                                               (11111111)  (51111)
                                                           (222111)
                                                           (321111)
                                                           (411111)
                                                           (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)

Mathematica

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

Crossrefs

The complement for equal differences is A049988, ranks A325328.

The complement is counted by A325325, ranks A325368.

The complement for compositions is A325545, ranks A389597.

These partitions are ranked by A389599.

For equal differences of compositions we have A389741, ranks A389735.

For compositions instead of partitions we have A389743, ranks A389598.

For equal instead of distinct differences we have A389811, ranks A389812.

A000041 counts integer partitions, strict A000009.

A000837 counts aperiodic partitions.

Cf. A007862, A051336, A175342, A325546, A332834, A389742, A389920.

Sequence in context: A174000 A241720 A224082 * A051678 A079906 A226814

Adjacent sequences: A389916 A389917 A389918 * A389920 A389921 A389922

Keyword

nonn

Author

Gus Wiseman, Oct 21 2025

Status

approved