A115624 - OEIS

%I #7 Jun 13 2020 00:49:24

%S 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,

%T 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,

%U 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

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

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

%H Robert Price, <a href="/A115624/b115624.txt">Table of n, a(n) for n = 1..9295</a> (first 25 rows).

%e 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.

%t sig[x_] := Length@NestWhileList[Last@Transpose@Tally@# &, x, # != {1} &, 1] - 1;

%t Table[sig /@ IntegerPartitions[n], {n, 8}]  // Flatten (* _Robert Price_, Jun 12 2020 *)

%Y 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.

%K easy,nonn

%O 1,3

%A _Franklin T. Adams-Watters_, Jan 25 2006