A325534 Number of separable partitions of n; see Comments.

1, 1, 1, 2, 3, 5, 6, 10, 14, 19, 26, 37, 49, 66, 87, 116, 152, 198, 254, 329, 422, 536, 678, 858, 1077, 1349, 1681, 2089, 2587, 3193, 3927, 4820, 5897, 7191, 8749, 10623, 12861, 15535, 18724, 22518, 27029, 32373, 38697, 46174, 54998, 65382, 77601, 91950, 108777

Offset: 0

Author

Clark Kimberling, May 08 2019

Keywords

• nonn
• easy

Comments

Definition: a partition is separable if there is an ordering of its parts in which no consecutive parts are identical; otherwise the partition is inseparable.

A partition with k parts is separable if and only if there is no part whose multiplicity is greater than ceiling(k/2). - Andrew Howroyd, Jan 31 2024

Examples

			For n=5, the partition 1+2+2 is separable as 2+1+2, and 2+1+1+1 is inseparable.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000041, A238589, A325535.

Programs

Mathematica

Table[Length[Select[Map[Quotient[(1 + Length[#]), Max[Map[Length, Split[#]]]] &,
IntegerPartitions[nn]], # > 1 &]], {nn, 50}]  (* Peter J. C. Moses, May 07 2019 *)

Formula

a(n) = A000041(n) - A325535(n).

Extensions

a(0)=1 prepended by Alois P. Heinz, Jan 20 2024