A325534 - OEIS

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A325534

Number of separable partitions of n; see Comments.

113

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

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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.