OFFSET
1,2
COMMENTS
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.
Also see Circular shift on Wikipedia.
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).
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.
The LCM of all cycle lengths gives A051732(n+1). - Alois P. Heinz, Apr 08 2020
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..3000
Wikipedia, Circular shift
FORMULA
a(n) = n <=> n in { A163782 } union { 1 }. - Alois P. Heinz, Apr 08 2020
EXAMPLE
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.
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.
PROG
(PARI)
Follow(s, f)={my(t=f(s), k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
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}
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Richard Locke Peterson, Mar 22 2020
EXTENSIONS
Terms a(20) and beyond from Andrew Howroyd, Mar 27 2020
STATUS
approved