A383709 Number of integer partitions of n with distinct multiplicities (Wilf) and distinct 0-appended differences.

1, 1, 2, 1, 2, 2, 3, 4, 4, 4, 5, 6, 5, 7, 8, 6, 8, 9, 9, 10, 9, 10, 12, 12, 11, 12, 14, 13, 14, 15, 14, 16, 16, 16, 18, 17, 17, 19, 20, 19, 19, 21, 21, 22, 22, 21, 24, 24, 23, 25, 25, 25, 26, 27, 27, 27, 28, 28, 30, 30, 28, 31, 32, 31, 32, 32, 33, 34, 34, 34

Offset: 0

Author

Gus Wiseman, May 15 2025

Keywords

• nonn

Comments

Integer partitions with distinct multiplicities are called Wilf partitions.

Examples

			The a(1) = 1 through a(8) = 4 partitions:
  (1)  (2)    (3)  (4)    (5)      (6)      (7)      (8)
       (1,1)       (2,2)  (3,1,1)  (3,3)    (3,2,2)  (4,4)
                                   (4,1,1)  (3,3,1)  (3,3,2)
                                            (5,1,1)  (6,1,1)
		

Crossrefs

For just distinct multiplicities we have A098859, ranks A130091, conjugate A383512.

For just distinct 0-appended differences we have A325324, ranks A325367.

For positive differences we have A383507, ranks A383532.

These partitions are ranked by A383712.

A048767 is the Look-and-Say transform, union A351294, complement A351295.

A239455 counts Look-and-Say partitions, complement A351293.

A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.

A383530 counts partitions that are not Wilf or conjugate-Wilf, ranks A383531.

A383534 gives 0-prepended differences by rank, see A325351.

Cf. A047966, A320348, A325325, A325349, A325368, A325388, A381431, A383506.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#]&&UnsameQ@@Differences[Append[#,0]]&]],{n,0,30}]

Formula

Ranked by A130091 /\ A325367