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

1,3

Comments

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

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].

Example

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.

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 *)

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.

Sequence in context: A256440 A088370 A328719 * A115624 A076291 A275015

Adjacent sequences: A113784 A113785 A113786 * A113788 A113789 A113790

Keyword

easy,nonn

Author

Franklin T. Adams-Watters, Jan 20 2006

Status

approved