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

%I A113787 #14 Jun 13 2020 00:49:13
%S A113787 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,
%T A113787 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,
%U A113787 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
%N A113787 Number of iterations of signature function required to get to [1] from partitions in Abramowitz and Stegun order.
%C A113787 The signature function takes a partition to the partition consisting of its repetition factors.
%H A113787 Robert Price, <a href="/A113787/b113787.txt">Table of n, a(n) for n = 1..9295</a> (first 25 rows).
%H A113787 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%e A113787 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.
%t A113787 sig[x_] := Length@NestWhileList[Last@Transpose@Tally@# &, x, # != {1} &, 1] - 1;
%t A113787 Table[sig /@ Sort[Reverse /@ IntegerPartitions[n]], {n, 9}]  // Flatten (* _Robert Price_, Jun 12 2020 *)
%Y A113787 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.
%K A113787 easy,nonn
%O A113787 1,3
%A A113787 _Franklin T. Adams-Watters_, Jan 20 2006