A113787 Number of iterations of signature function required to get to [1] from partitions in Abramowitz and Stegun order.

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

Offset: 1

Author

Franklin T. Adams-Watters, Jan 20 2006

Keywords

• easy
• nonn

Comments

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

Examples

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

Links

• Robert Price, Table of n, a(n) for n = 1..9295 (first 25 rows).
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Crossrefs

Cf. A115621, A115624, Sequence of first partitions with a(m)=n is A012257, with initial rows {1} and {2} in prepended. See A036036 for A&S partitions.

Programs

Mathematica

sig[x_] := Length@NestWhileList[Last@Transpose@Tally@# &, x, # != {1} &, 1] - 1;
Table[sig /@ Sort[Reverse /@ IntegerPartitions[n]], {n, 9}]  // Flatten (* Robert Price, Jun 12 2020 *)