A355393 - OEIS

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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

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OFFSET

0,4

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.

LINKS

Table of n, a(n) for n=0..50.

EXAMPLE

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)

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}]

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.

Sequence in context: A237752 A032480 A226137 * A163771 A194855 A272766

Adjacent sequences: A355390 A355391 A355392 * A355394 A355395 A355396

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 26 2022

STATUS

approved