A355393 Number of integer partitions of n such that, for all parts x of multiplicity 1, either x - 1 or x + 1 is also a part.

1, 0, 1, 2, 3, 4, 6, 7, 10, 14, 17, 23, 32, 39, 51, 67, 83, 105, 134, 165, 206, 256, 312, 385, 475, 573, 697, 849, 1021, 1231, 1483, 1771, 2121, 2534, 3007, 3575, 4245, 5008, 5914, 6979, 8198, 9626, 11292, 13201, 15430, 18010, 20960, 24389, 28346, 32855, 38066

Offset: 0

Author

Gus Wiseman, Aug 26 2022

Keywords

• nonn

Comments

These are partitions without a neighborless singleton, where a part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once.

Examples

			The a(0) = 1 through a(8) = 10 partitions:
  ()  .  (11)  (21)   (22)    (32)     (33)      (43)       (44)
               (111)  (211)   (221)    (222)     (322)      (332)
                      (1111)  (2111)   (321)     (2221)     (2222)
                              (11111)  (2211)    (3211)     (3221)
                                       (21111)   (22111)    (3311)
                                       (111111)  (211111)   (22211)
                                                 (1111111)  (32111)
                                                            (221111)
                                                            (2111111)
                                                            (11111111)
		

Crossrefs

This is the singleton case of A355394, complement A356236.

The complement is counted by A356235.

These partitions are ranked by the complement of A356237.

The strict case is A356606, complement A356607.

A000041 counts integer partitions, strict A000009.

A000837 counts relatively prime partitions, ranked by A289509.

A007690 counts partitions with no singletons, complement A183558.

Cf. A073491, A325160, A328171, A328172, A328187, A328220, A356233.

Programs

Mathematica

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