A326370 Number of condensations to convert all the partitions of n to strict partitions of n.

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

Offset: 1

Author

Clark Kimberling, Jul 06 2019

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

• Rémy Sigrist, PARI program for A326370

Crossrefs

Cf. A000009, A000041, A239312, A326371.

Programs

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}]

PARI

See Links section.

Extensions

More terms from Rémy Sigrist, Jul 07 2019