This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A186144 #26 Dec 12 2024 09:43:27 %S A186144 1,1,3,3,2,1,2,1,2,1,2,1,2 %N 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). %C A186144 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). %C A186144 From _Ben Whitmore_, Nov 14 2020: (Start) %C A186144 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 %C A186144 [2, 1, (n+3)/2, n, n-1, ..., (n+5)/2, (n+1)/2, (n-1)/2, ..., 4, 3], %C A186144 [2, 1, n-1, n-2, ..., (n+3)/2, n, (n+1)/2, (n-1)/2, ..., 4, 3], %C A186144 and for even n>5, we have a(n) = 1 and the action of the permutation is %C A186144 [2, n, 1, n-1, n-2, ..., 4, 3]. %C A186144 (End) %F A186144 Conjecture: For n>4, a(n) = 1 if n is even, a(n) = 2 if n is odd. - _Ben Whitmore_, Nov 14 2020 %Y A186144 Cf. A039745, A378881. %K A186144 nonn,hard,more %O A186144 1,3 %A A186144 _Tony Bartoletti_, Feb 23 2011 %E A186144 a(10)-a(13) by _Ben Whitmore_, Nov 14 2020