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

0,4

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

Links

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

Formula

a(n) + A325535(n) = A000041(n) = number of partitions of n.

Example

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

Mathematica

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

Crossrefs

Cf. A000041, A238589, A325535.

Sequence in context: A325713 A325714 A325715 * A280013 A039840 A039845

Adjacent sequences: A325531 A325532 A325533 * A325535 A325536 A325537

Keyword

nonn,easy

Author

Clark Kimberling, May 08 2019

Extensions

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

Status

approved