A356235 - OEIS

%I #7 Aug 25 2022 08:33:36

%S 0,1,1,1,2,3,5,8,12,16,25,33,45,62,84,109,148,192,251,325,421,536,690,

%T 870,1100,1385,1739,2161,2697,3334,4121,5071,6228,7609,9303,11308,

%U 13732,16629,20101,24206,29140,34957,41882,50060,59745,71124,84598,100365

%N Number of integer partitions of n with a neighborless singleton.

%C A part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once. Examples of partitions with a neighborless singleton are: (3), (3,1), (3,1,1), (3,3,1). Examples of partitions without a neighborless singleton are: (3,3,1,1), (4,3,1,1), (3,2,1), (2,1), (3,3).

%e The a(1) = 1 through a(8) = 12 partitions:

%e   (1)  (2)  (3)  (4)   (5)    (6)     (7)      (8)

%e                  (31)  (41)   (42)    (52)     (53)

%e                        (311)  (51)    (61)     (62)

%e                               (411)   (331)    (71)

%e                               (3111)  (421)    (422)

%e                                       (511)    (431)

%e                                       (4111)   (521)

%e                                       (31111)  (611)

%e                                                (4211)

%e                                                (5111)

%e                                                (41111)

%e                                                (311111)

%t Table[Length[Select[IntegerPartitions[n],Min@@Length/@Split[Reverse[#],#1>=#2-1&]==1&]],{n,0,30}]

%Y The complement is counted by A355393.

%Y This is the singleton case of A356236, complement A355394.

%Y These partitions are ranked by A356237.

%Y The strict case is A356607, complement A356606.

%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, A325160, A328171, A328172, A328187, A328221, A356233.

%K nonn

%O 0,5

%A _Gus Wiseman_, Aug 23 2022