A115624 Number of iterations of signature function required to get to [1] from partitions in Mathematica order.

0, 1, 2, 1, 3, 2, 1, 3, 2, 4, 2, 1, 3, 3, 4, 4, 4, 2, 1, 3, 3, 4, 2, 3, 4, 2, 3, 4, 2, 1, 3, 3, 4, 3, 3, 4, 4, 4, 5, 4, 4, 4, 4, 2, 1, 3, 3, 4, 3, 3, 4, 2, 3, 4, 5, 4, 4, 3, 5, 5, 4, 2, 4, 4, 4, 2, 1, 3, 3, 4, 3, 3, 4, 3, 3, 4, 5, 4, 4, 3, 5, 5, 5, 4, 2, 5, 4, 4, 5, 5, 4, 4, 3, 4, 4, 2

Offset: 1

Author

Franklin T. Adams-Watters, Jan 25 2006

Keywords

• easy
• nonn

Comments

The signature function takes a partition to the partition consisting of its repetition factors.

Examples

			Partition 5 in Mathematica order is [2,1]. Applying the signature function to this repeatedly gives [2,1] -> [1^2] -> [2] -> [1], so a(5)=3.
		

Links

• Robert Price, Table of n, a(n) for n = 1..9295 (first 25 rows).

Crossrefs

Cf. A115621, A113787, Sequence of first partitions with a(m)=n is A012257, with initial rows {1} and {2} in prepended. See A080577 for Mathematica partition order.

Programs

Mathematica

sig[x_] := Length@NestWhileList[Last@Transpose@Tally@# &, x, # != {1} &, 1] - 1;
Table[sig /@ IntegerPartitions[n], {n, 8}]  // Flatten (* Robert Price, Jun 12 2020 *)