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A333468 Length of the largest disjoint cycle of the permutation that results from the composition of first n circular shifts.

%I A333468 #41 Apr 09 2020 17:32:34
%S A333468 1,2,2,3,5,6,3,4,9,4,7,10,9,14,4,5,7,18,8,10,7,7,14,11,6,26,12,9,29,
%T A333468 30,5,6,33,11,21,6,11,15,22,27,41,6,17,8,8,7,22,24,15,50,28,8,53,18,
%U A333468 22,14,25,9,15,55,14,50,6,7,65,11,19,34,69,23,35,14,22,74,10
%N A333468 Length of the largest disjoint cycle of the permutation that results from the composition of first n circular shifts.
%C A333468 Size of the largest part of the partition of n that is associated with the cycle structure of the permutation given by the permutation product (1)*(1,2)*(1,2,3)*...*(1,2,3,...n) after the product is rewritten as the product of disjoint cycles, where * means functional composition, and the permutations are written in cycle form.
%C A333468 Also see Circular shift on Wikipedia.
%C A333468 For n>1, a(n) is always greater than 1, since the given product can never be the identity permutation on the set {1,2,...,n}, which is the only permutation associated with the partition <1,1,...,1> (1 repeated n times).
%C A333468 Connections: The image of 1 in each resulting permutation appears to be the same as the numbers in A003602. The number of parts in the partition associated with each resulting permutation appear to match the numbers in A006694.
%C A333468 The LCM of all cycle lengths gives A051732(n+1). - _Alois P. Heinz_, Apr 08 2020
%H A333468 Alois P. Heinz, <a href="/A333468/b333468.txt">Table of n, a(n) for n = 1..3000</a>
%H A333468 Wikipedia, <a href="https://en.wikipedia.org/wiki/Circular_shift">Circular shift</a>
%F A333468 a(n) = n <=> n in { A163782 } union { 1 }. - _Alois P. Heinz_, Apr 08 2020
%e A333468 For n=3, the permutation (1)*(1,2)*(1,2,3)=(1)*(2,3), which is associated with the partition <2,1> of 3. The size of the largest part is 2, so a(3)=2.
%e A333468 For n=11, the permutation (1)*(1,2)*..*(1,2,..11)=(1,2,7,5)*(3,4,8,10,11,6,9) when rewritten as the product of disjoint cycles, which is associated with the partition <7,4> of 11. The size of the largest part is 7, so a(11)=7.
%o A333468 (PARI)
%o A333468 Follow(s, f)={my(t=f(s), k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
%o A333468 mkp(n)={my(v=vector(n,i,i)); for(k=1, n, my(t=v[1]); for(i=1, k-1, v[i]=v[i+1]); v[k]=t); v}
%o A333468 a(n)={my(v=mkp(n), m=0); for(i=1, n, m=max(m, Follow(i, j->v[j]))); m} \\ _Andrew Howroyd_, Mar 27 2020
%Y A333468 Cf. A003602, A006694, A051732, A071642, A163782.
%K A333468 nonn
%O A333468 1,2
%A A333468 _Richard Locke Peterson_, Mar 22 2020
%E A333468 Terms a(20) and beyond from _Andrew Howroyd_, Mar 27 2020