

A326370


Number of condensations to convert all the partitions of n to strict partitions of n.


2



0, 1, 1, 2, 1, 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6
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OFFSET

1,4


COMMENTS

Suppose that p is a partition of n. Let x(1), x(2), ..., x(k) be the distinct parts of p, and let m(i) be the multiplicity of x(i) in p. The partition {m(1)*x(1), m(2)*x(2), ..., x(k)*m(k)} of n is called the condensation of p.


LINKS

Table of n, a(n) for n=1..87.
Rémy Sigrist, PARI program for A326370


EXAMPLE

The condensation of [4, 2, 1, 1] is [4, 2, 2], of which the condensation is [4, 4], of which condensation is [8]; thus, a total of three condensations. This is maximal for the partitions of 8, so that a(8) = 3. See A239312.


MATHEMATICA

f[m_] := Table[Tally[m][[h]][[1]]*Tally[m][[h]][[2]], {h, 1, Length[Tally[m]]}];
m[n_, k_] := IntegerPartitions[n][[k]];
q[n_, k_] := 2 + Length[FixedPointList[f, m[n, k]]];
a[n_] := Max[Table[q[n, k], {k, 1, PartitionsP[n]}]];
Table[a[n], {n, 1, 30}]


PROG

(PARI) See Links section.


CROSSREFS

Cf. A000009, A000041, A239312, A326371.
Sequence in context: A071068 A240872 A328806 * A137735 A290983 A272887
Adjacent sequences: A326367 A326368 A326369 * A326371 A326372 A326373


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Jul 06 2019


EXTENSIONS

More terms from Rémy Sigrist, Jul 07 2019


STATUS

approved



