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 A333468 Length of the largest disjoint cycle of the permutation that results from the composition of first n circular shifts. 1
 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, 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, 22, 14, 25, 9, 15, 55, 14, 50, 6, 7, 65, 11, 19, 34, 69, 23, 35, 14, 22, 74, 10 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A003602, A006694, A051732, A071642, A163782. Sequence in context: A101779 A169891 A137756 * A364971 A225489 A308773 Adjacent sequences: A333465 A333466 A333467 * A333469 A333470 A333471 KEYWORD nonn AUTHOR Richard Locke Peterson, Mar 22 2020 EXTENSIONS Terms a(20) and beyond from Andrew Howroyd, Mar 27 2020 STATUS approved

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Last modified September 23 03:44 EDT 2023. Contains 365532 sequences. (Running on oeis4.)