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Number of integer partitions of n such that, for all parts x, x - 1 or x + 1 is also a part.
11

%I #24 Feb 17 2024 14:08:02

%S 1,0,0,1,1,3,3,6,6,10,11,16,18,25,30,38,47,59,74,90,112,136,171,203,

%T 253,299,372,438,536,631,767,900,1085,1271,1521,1774,2112,2463,2910,

%U 3389,3977,4627,5408,6276,7304,8459,9808,11338,13099,15112,17404,20044,23018,26450,30299,34746,39711,45452,51832

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

%C These are partitions without a neighborless part, where a part x is neighborless if neither x - 1 nor x + 1 are parts. The first counted partition that does not cover an interval is (5,4,2,1).

%H Lucas A. Brown, <a href="/A355394/b355394.txt">Table of n, a(n) for n = 0..100</a>

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A355394.py">A355394.py</a>

%F a(n) = A000041(n) - A356236(n).

%e The a(0) = 1 through a(9) = 11 partitions:

%e () . . (21) (211) (32) (321) (43) (332) (54)

%e (221) (2211) (322) (3221) (432)

%e (2111) (21111) (2221) (22211) (3222)

%e (3211) (32111) (3321)

%e (22111) (221111) (22221)

%e (211111) (2111111) (32211)

%e (222111)

%e (321111)

%e (2211111)

%e (21111111)

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

%Y The singleton case is A355393, complement A356235.

%Y The complement is counted by A356236, ranked by A356734.

%Y The strict case is A356606, complement A356607.

%Y These partitions are ranked by A356736.

%Y A000041 counts integer partitions, strict A000009.

%Y A000837 counts relatively prime partitions, ranked by A289509.

%Y A007690 counts partitions with no singletons, complement A183558.

%Y Cf. A066312, A073491, A077855, A328171, A328172, A328187, A328221, A356233, A356237.

%K nonn

%O 0,6

%A _Gus Wiseman_, Aug 26 2022

%E a(31)-a(59) from _Lucas A. Brown_, Sep 04 2022