A364674 Number of integer partitions of n containing all of their own nonzero first differences.

1, 1, 2, 3, 4, 4, 8, 7, 11, 13, 17, 18, 32, 30, 44, 54, 70, 78, 114, 125, 171, 205, 257, 302, 408, 464, 592, 711, 892, 1042, 1330, 1543, 1925, 2279, 2787, 3291, 4061, 4727, 5753, 6792, 8197, 9583, 11593, 13505, 16198, 18965, 22548, 26290, 31340, 36363, 43046

Offset

0,3

Links

Table of n, a(n) for n=0..50.

Example

The partition (10,5,3,3,2,1) has nonzero differences (5,2,1,1) so is counted under a(24).
The a(1) = 1 through a(9) = 13 partitions:
  (1) (2)  (3)   (4)    (5)     (6)      (7)       (8)        (9)
      (11) (21)  (22)   (221)   (33)     (421)     (44)       (63)
           (111) (211)  (2111)  (42)     (2221)    (422)      (333)
                 (1111) (11111) (222)    (3211)    (2222)     (3321)
                                (321)    (22111)   (3221)     (4221)
                                (2211)   (211111)  (4211)     (22221)
                                (21111)  (1111111) (22211)    (32211)
                                (111111)           (32111)    (42111)
                                                   (221111)   (222111)
                                                   (2111111)  (321111)
                                                   (11111111) (2211111)
                                                              (21111111)
                                                              (111111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], SubsetQ[#, Differences[Union[#]]]&]], {n, 0, 30}]

Crossrefs

For no differences we have A363260, subsets A364463, strict A364464.

For at least one difference we have A364467, ranks A364537, strict A364536.

For subsets instead of partitions we have A364671, complement A364672.

The strict case (no differences of 0) is counted by A364673.

For submultisets instead of subsets we have A364675.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A236912 counts sum-free partitions w/o re-used parts, complement A237113.

A325325 counts partitions with distinct first differences.

Cf. A002865, A007862, A025065, A229816, A237667, A320347, A326083, A363225, A364272, A364466.

Sequence in context: A317085 A236543 A360245 * A319079 A325329 A224038

Adjacent sequences: A364671 A364672 A364673 * A364675 A364676 A364677

Keyword

nonn

Author

Gus Wiseman, Aug 04 2023

Status

approved