A356236 Number of integer partitions of n with a neighborless part.

0, 1, 2, 2, 4, 4, 8, 9, 16, 20, 31, 40, 59, 76, 105, 138, 184, 238, 311, 400, 515, 656, 831, 1052, 1322, 1659, 2064, 2572, 3182, 3934, 4837, 5942, 7264, 8872, 10789, 13109, 15865, 19174, 23105, 27796, 33361, 39956, 47766, 56985, 67871, 80675, 95750, 113416

Offset

0,3

Comments

A part x of a partition is neighborless if neither x - 1 nor x + 1 are parts.

Links

Table of n, a(n) for n=0..47.

Formula

a(n) = A000041(n) - A355394(n).

Example

The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (41)     (33)      (52)
                    (31)    (311)    (42)      (61)
                    (1111)  (11111)  (51)      (331)
                                     (222)     (421)
                                     (411)     (511)
                                     (3111)    (4111)
                                     (111111)  (31111)
                                               (1111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], Function[ptn, Or@@Table[!MemberQ[ptn, x-1]&&!MemberQ[ptn, x+1], {x, Union[ptn]}]]]], {n, 0, 30}]

Crossrefs

The complement is counted by A355394, singleton case A355393.

The singleton case is A356235, ranked by A356237.

The strict case is A356607, complement A356606.

These partitions are ranked by the complement of A356736.

A000041 counts integer partitions, strict A000009.

A000837 counts relatively prime partitions, ranked by A289509.

A007690 counts partitions with no singletons, complement A183558.

Cf. A066205, A112798, A319630, A325160, A328171, A328172, A328187, A328221.

Sequence in context: A222708 A324843 A306692 * A120803 A316624 A318770

Adjacent sequences: A356233 A356234 A356235 * A356237 A356238 A356239

Keyword

nonn

Author

Gus Wiseman, Aug 24 2022

Status

approved