A186144 Number of elements in the symmetric group S_n whose distance from a fixed element is the group diameter under compositions of (1,2) and (1,2,...,n).

1, 1, 3, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 1

Author

Tony Bartoletti, Feb 23 2011

Keywords

• nonn
• hard
• more

Comments

a(n) is the number of elements in the symmetric group S_n that are maximally distant from any fixed element, where distance is taken to be the minimal sequence of operations composed from transposition (1,2) and rotation (1,2,...,n) producing one element from another. This maximal distance is the diameter of S_n under the stated compositions, given by A039745(n).

From Ben Whitmore, Nov 14 2020: (Start)

Conjecture (verified up to n = 13): Consider the a(n) permutations that take A039745(n) steps to reach the identity. For odd n>5, we have a(n) = 2 and the actions of these permutations on the list [1, 2, ..., n] are

[2, 1, (n+3)/2, n, n-1, ..., (n+5)/2, (n+1)/2, (n-1)/2, ..., 4, 3],

[2, 1, n-1, n-2, ..., (n+3)/2, n, (n+1)/2, (n-1)/2, ..., 4, 3],

and for even n>5, we have a(n) = 1 and the action of the permutation is

[2, n, 1, n-1, n-2, ..., 4, 3].

(End)

Crossrefs

Cf. A039745, A378881.

Formula

Conjecture: For n>4, a(n) = 1 if n is even, a(n) = 2 if n is odd. - Ben Whitmore, Nov 14 2020

Extensions

a(10)-a(13) by Ben Whitmore, Nov 14 2020