A383507 Number of Wilf and conjugate Wilf integer partitions of n.

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 20, 27, 30, 31, 40, 50, 56, 68, 76, 86, 112, 126, 139, 170, 197, 216, 251, 297, 317, 378, 411, 466, 521, 607, 621, 745, 791, 892, 975, 1123, 1163, 1366, 1439, 1635, 1757, 2021, 2080, 2464, 2599, 2882, 3116, 3572, 3713

Offset

0,3

Comments

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

Links

Table of n, a(n) for n=0..54.

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

Formula

These partitions have Heinz numbers A130091 /\ A383512.

Example

The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (1111)  (11111)  (222)     (331)      (332)
                                     (411)     (511)      (611)
                                     (3111)    (4111)     (2222)
                                     (111111)  (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)

Mathematica

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

Crossrefs

A048768 gives Look-and-Say fixed points, counted by A217605.

A098859 counts Wilf partitions, ranks A130091, conjugate A383512.

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

A325349 counts partitions with distinct augmented differences, ranks A325366.

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

A381431 is the section-sum transform, union A381432, complement A381433.

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

A383709 counts Wilf partitions with distinct 0-appended differences.

Cf. A047966, A048767, A111133, A320348, A325324, A325325, A325367, A325368, A325388, A351294, A383506.

Sequence in context: A389802 A054172 A236971 * A383519 A383508 A373446

Adjacent sequences: A383504 A383505 A383506 * A383508 A383509 A383510

Keyword

nonn

Author

Gus Wiseman, May 14 2025

Status

approved