A356235 Number of integer partitions of n with a neighborless singleton.

0, 1, 1, 1, 2, 3, 5, 8, 12, 16, 25, 33, 45, 62, 84, 109, 148, 192, 251, 325, 421, 536, 690, 870, 1100, 1385, 1739, 2161, 2697, 3334, 4121, 5071, 6228, 7609, 9303, 11308, 13732, 16629, 20101, 24206, 29140, 34957, 41882, 50060, 59745, 71124, 84598, 100365

Offset: 0

Author

Gus Wiseman, Aug 23 2022

Keywords

• nonn

Comments

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).

Examples

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)  (3)  (4)   (5)    (6)     (7)      (8)
                 (31)  (41)   (42)    (52)     (53)
                       (311)  (51)    (61)     (62)
                              (411)   (331)    (71)
                              (3111)  (421)    (422)
                                      (511)    (431)
                                      (4111)   (521)
                                      (31111)  (611)
                                               (4211)
                                               (5111)
                                               (41111)
                                               (311111)
		

Crossrefs

The complement is counted by A355393.

This is the singleton case of A356236, complement A355394.

These partitions are ranked by A356237.

The strict case is A356607, complement A356606.

A000041 counts integer partitions, strict A000009.

A000837 counts relatively prime partitions, ranked by A289509.

A007690 counts partitions with no singletons, complement A183558.

Cf. A066205, A325160, A328171, A328172, A328187, A328221, A356233.

Programs

Mathematica

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