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%I #68 Sep 10 2020 13:59:49
%S 0,1,2,6,10,15,21,28,36,45,55,66,78
%N Diameter of the symmetric group S_n when generated by the transposition (1,2) and both left and right rotations by (1,2,...,n).
%C Given an ordered sequence of n elements (1,2,3,...,n), let X represent the permutation that transposes the first two elements, X(1,2,3,...,n) = (2,1,3,...,n), let L be the "left rotation" of the sequence, L(1,2,3,...,n) = (2,3,...,n,1), and let R be the "right rotation", R(1,2,3,...,n) = (n,1,2,...,n-1). Then every permutation of (1,2,3,...,n) can be expressed as a composition of the permutations X, L and R. One can exhaustively generate such compositions by taking L="0", X="1", R="2", and considering, in turn, base 3 numbers of increasing length (padded with leading zeros). Note that any base 3 number containing the subsequence "11", "02" or "20" may be discarded.
%C Note also that by defining the distance between any two permutations p and q in S_n, dist(p,q), to be the length of the minimal composition of LXR transforming p into q, we have dist(p,q) = dist(q,p), owing to L and R being mutually inverse, and X being self-inverse.
%H Danilo Bazzanella, Antonio Di Scala, Simone Dutto, Nadir Murru, <a href="https://arxiv.org/abs/2002.08062">Primality tests, linear recurrent sequences and the Pell equation</a>, arXiv:2002.08062 [math.NT], 2020.
%H Sai Satwik Kuppili, Bhadrachalam Chitturi, <a href="https://arxiv.org/abs/2002.07342">An Upper Bound for Sorting R_n with LRE</a>, arXiv:2002.07342 [cs.DS], 2020.
%F Conjecture: a(n) = - Sum_{k=1..n-1} Stirling1(n+k-1, (n-1)*k). This formula holds for all known n. - _Arkadiusz Wesolowski_, Mar 30 2013. For n>3, this formula contains only one nonzero term (for k=1) and reduces to the formula n*(n-1)/2 conjectured below. - _Max Alekseyev_, Sep 10 2020
%F Conjecture: a(n) = n*(n-1)/2 for all n>3. - _Per W. Alexandersson_, Aug 25 2020
%F Conjectures from _Colin Barker_, Aug 26 2020: (Start)
%F G.f.: x^2*(1 - x + 3*x^2 - 3*x^3 + x^4) / (1 - x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F (End)
%e The diameter of S_5 is 10, given this set of generators, since there is no sequence shorter than 0010010121 (i.e., LLXLLXLXRX) that will transform (1,2,3,4,5) into (2,1,5,4,3), and there is no permutation of (1,2,3,4,5) that requires more than a length-10 composition of L, X and R. Thus a(5) = 10.
%t a[1] = 0; a[n_] := GraphDiameter[CayleyGraph[PermutationGroup[{Cycles[{{1, 2}}], Cycles[{Range[n]}], InversePermutation[Cycles[{Range[n]}]]}]]]; (* _Ben Whitmore_, Jan 09 2018 *)
%o (Sage) def a(n): return PermutationGroup([[(1, 2)], [tuple(1..n)], PermutationGroupElement([tuple(1..n)])^(-1)]).cayley_graph().diameter() # _Max Alekseyev_
%Y Cf. A039745, A186752, A000217.
%K nonn,more,hard
%O 1,3
%A _Tony Bartoletti_, Feb 26 2011
%E a(8)=28 added by _Tony Bartoletti_, Mar 12 2011
%E a(9)=36 added by _R. H. Hardin_, Sep 09 2011
%E a(10)=45 added by _Sharon Li_, Mar 09 2013
%E a(11)=55 and a(12)=66 added by _James Bieron_, Mar 15 2013
%E a(13)=78 added by _Ben Whitmore_, Jan 09 2018
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