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A325535 Number of inseparable partitions of n; see Comments. 102
0, 1, 1, 2, 2, 5, 5, 8, 11, 16, 19, 28, 35, 48, 60, 79, 99, 131, 161, 205, 256, 324, 397, 498, 609, 755, 921, 1131, 1372, 1677, 2022, 2452, 2952, 3561, 4260, 5116, 6102, 7291, 8667, 10309, 12210, 14477, 17087, 20177, 23752, 27957, 32804, 38496, 45049, 52704 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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.

LINKS

Table of n, a(n) for n=1..50.

FORMULA

a(n) + A325534(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.

From Gus Wiseman, Jun 27 2020: (Start)

The a(2) = 2 through a(9) = 11 inseparable partitions:

  11   111   22     2111    33       2221      44         333

             1111   11111   222      4111      2222       3222

                            3111     31111     5111       6111

                            21111    211111    41111      22221

                            111111   1111111   221111     51111

                                               311111     321111

                                               2111111    411111

                                               11111111   2211111

                                                          3111111

                                                          21111111

                                                          111111111

(End)

MATHEMATICA

u=Table[Length[Select[Map[Quotient[(1 + Length[#]), Max[Map[Length, Split[#]]]] &,

IntegerPartitions[nn]], # > 1 &]], {nn, 50}]

Table[PartitionsP[n] - u[[n]], {n, 1, Length[u]}]

(* Peter J. C. Moses, May 07 2019 *)

Table[Length[Select[IntegerPartitions[n], Select[Permutations[#], !MatchQ[#, {___, x_, x_, ___}]&]=={}&]], {n, 10}] (* Gus Wiseman, Jun 27 2020 *)

CROSSREFS

The Heinz numbers of these partitions are given by A335448.

Strict partitions are counted by A000009 and are all separable.

Anti-run compositions are counted by A003242.

Anti-run patterns are counted by A005649.

Partitions whose differences are an anti-run are A238424.

Separable partitions are counted by A325534.

Anti-run compositions are ranked by A333489.

Anti-run permutations of prime indices are counted by A335452.

Cf. A000041, A106356, A261962, A292884, A332668, A333175.

Sequence in context: A222706 A240495 A304393 * A345165 A062405 A071181

Adjacent sequences:  A325532 A325533 A325534 * A325536 A325537 A325538

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 08 2019

STATUS

approved

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Last modified May 19 17:44 EDT 2022. Contains 353847 sequences. (Running on oeis4.)