A039745 Diameter of symmetric group S_n when generated by (1,2) and (1,2,3,...,n).

0, 1, 2, 6, 11, 18, 25, 35, 45, 58, 71, 87, 103

Offset

1,3

Comments

a(n) is smallest number such that every element of S_n can be written as a product of at most a(n) terms each of which is the transposition (1,2) or the n-cycle (1,2,3,...,n).

The distinction between A039745 (this sequence) and A186783 comes from whether we treat the Cayley graph of the generating set as directed or undirected (alternatively, whether we allow multiplication by inverses of generators when constructing elements). A039745 deals with the directed Cayley graph, while A186783 deals with the undirected one. - Max Alekseyev, Sep 09 2011

Links

Table of n, a(n) for n=1..13.

Example

a(3)=2 because (1,3,2) = (1,2,3)(1,2).

Mathematica

a[n_] := GraphDiameter[CayleyGraph[SymmetricGroup[n]]] (* Ben Whitmore, Nov 13 2020 *)

Prog

(Sage) def a(n): return PermutationGroup([[(1, 2)], [tuple(1..n)]]).cayley_graph().diameter() # Max Alekseyev, Mar 02 2010

Crossrefs

Cf. A378881 (antipodal permutations), A186144 (number of them).

Cf. A186783 (LRE diameter).

Sequence in context: A231559 A104813 A239698 * A298872 A298875 A329598

Adjacent sequences: A039742 A039743 A039744 * A039746 A039747 A039748

Keyword

hard,nonn,nice,more

Author

David desJardins

Extensions

a(12)-a(13) by Ben Whitmore, Nov 12 2020

Status

approved