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, 122, 141

Offset: 1

Author

David desJardins

Keywords

• hard
• nonn
• nice
• more

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

Examples

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

Links

• Dmytro Fedoriaka, Growth function for n=15.

Crossrefs

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

Cf. A186783 (LRE diameter).

Programs

Mathematica

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

Sage

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

Extensions

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

a(14) by Dmytro Fedoriaka, Jun 30 2025

a(15) by Dmytro Fedoriaka, Jul 14 2025