|
%I #11 Feb 17 2024 14:08:09
%S 0,1,2,2,4,4,8,9,16,20,31,40,59,76,105,138,184,238,311,400,515,656,
%T 831,1052,1322,1659,2064,2572,3182,3934,4837,5942,7264,8872,10789,
%U 13109,15865,19174,23105,27796,33361,39956,47766,56985,67871,80675,95750,113416
%N Number of integer partitions of n with a neighborless part.
%C A part x of a partition is neighborless if neither x - 1 nor x + 1 are parts.
%F a(n) = A000041(n) - A355394(n).
%e The a(1) = 1 through a(8) = 9 partitions:
%e (1) (2) (3) (4) (5) (6) (7)
%e (11) (111) (22) (41) (33) (52)
%e (31) (311) (42) (61)
%e (1111) (11111) (51) (331)
%e (222) (421)
%e (411) (511)
%e (3111) (4111)
%e (111111) (31111)
%e (1111111)
%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 complement is counted by A355394, singleton case A355393.
%Y The singleton case is A356235, ranked by A356237.
%Y The strict case is A356607, complement A356606.
%Y These partitions are ranked by the complement of 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. A066205, A112798, A319630, A325160, A328171, A328172, A328187, A328221.
%K nonn
%O 0,3
%A _Gus Wiseman_, Aug 24 2022
|
