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

Author

Gus Wiseman, Aug 04 2023

Keywords

• nonn

Examples

			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)
		

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.

Programs

Mathematica

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