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A355394 Number of integer partitions of n such that, for all parts x, x - 1 or x + 1 is also a part.

%I A355394 #24 Feb 17 2024 14:08:02
%S A355394 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 A355394 253,299,372,438,536,631,767,900,1085,1271,1521,1774,2112,2463,2910,
%U A355394 3389,3977,4627,5408,6276,7304,8459,9808,11338,13099,15112,17404,20044,23018,26450,30299,34746,39711,45452,51832
%N A355394 Number of integer partitions of n such that, for all parts x, x - 1 or x + 1 is also a part.
%C A355394 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 A355394 Lucas A. Brown, <a href="/A355394/b355394.txt">Table of n, a(n) for n = 0..100</a>
%H A355394 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A355394.py">A355394.py</a>
%F A355394 a(n) = A000041(n) - A356236(n).
%e A355394 The a(0) = 1 through a(9) = 11 partitions:
%e A355394   ()  .  .  (21)  (211)  (32)    (321)    (43)      (332)      (54)
%e A355394                          (221)   (2211)   (322)     (3221)     (432)
%e A355394                          (2111)  (21111)  (2221)    (22211)    (3222)
%e A355394                                           (3211)    (32111)    (3321)
%e A355394                                           (22111)   (221111)   (22221)
%e A355394                                           (211111)  (2111111)  (32211)
%e A355394                                                                (222111)
%e A355394                                                                (321111)
%e A355394                                                                (2211111)
%e A355394                                                                (21111111)
%t A355394 Table[Length[Select[IntegerPartitions[n],Function[ptn,!Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]
%Y A355394 The singleton case is A355393, complement A356235.
%Y A355394 The complement is counted by A356236, ranked by A356734.
%Y A355394 The strict case is A356606, complement A356607.
%Y A355394 These partitions are ranked by A356736.
%Y A355394 A000041 counts integer partitions, strict A000009.
%Y A355394 A000837 counts relatively prime partitions, ranked by A289509.
%Y A355394 A007690 counts partitions with no singletons, complement A183558.
%Y A355394 Cf. A066312, A073491, A077855, A328171, A328172, A328187, A328221, A356233, A356237.
%K A355394 nonn
%O A355394 0,6
%A A355394 _Gus Wiseman_, Aug 26 2022
%E A355394 a(31)-a(59) from _Lucas A. Brown_, Sep 04 2022