A356607 Number of strict integer partitions of n with at least one neighborless part.

0, 1, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 13, 17, 20, 24, 30, 36, 41, 52, 60, 71, 84, 100, 114, 137, 158, 183, 214, 248, 283, 330, 379, 432, 499, 570, 648, 742, 846, 955, 1092, 1234, 1395, 1580, 1786, 2005, 2270, 2548, 2861, 3216, 3610, 4032, 4526, 5055, 5642, 6304, 7031, 7820, 8720, 9694

Offset: 0

Author

Gus Wiseman, Aug 26 2022

Keywords

• nonn

Comments

A part x is neighborless if neither x - 1 nor x + 1 are parts.

Examples

			The a(0) = 0 through a(9) = 6 partitions:
  .  (1)  (2)  (3)  (4)   (5)   (6)   (7)    (8)    (9)
                    (31)  (41)  (42)  (52)   (53)   (63)
                                (51)  (61)   (62)   (72)
                                      (421)  (71)   (81)
                                             (431)  (531)
                                             (521)  (621)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 101 terms from Lucas A. Brown)
• Lucas A. Brown, A356607.py

Crossrefs

This is the strict case of A356235 and A356236.

The complement is counted by A356606, non-strict A355393 and A355394.

A000041 counts integer partitions, strict A000009.

A000837 counts relatively prime partitions, ranked by A289509.

A007690 counts partitions with no singletons, complement A183558.

Cf. A073492, A137921, A325160, A328171, A328172, A328187, A328220, A328221.

Programs

Mathematica

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

Extensions

a(31)-a(59) from Lucas A. Brown, Sep 09 2022