OFFSET
0,3
COMMENTS
From Paul Boddington, Feb 24 2015: (Start)
Suppose G is the symmetric group on n letters. For each g in G, the set of f satisfying fgg = ggf is just the centralizer Z_gg(G). However |Z_gg(G)| is clearly constant on conjugacy classes of G. By the orbit-stabilizer theorem the size of the conjugacy class containing g is |G| / |Z_g(G)|. Since |G| = n! and Z_g(G) is a subgroup of Z_gg(G) we see that a(n) equals n! multiplied by the sum of indices |Z_gg(G) : Z_g(G)| where the sum is over representatives of the conjugacy classes of G. Since the conjugacy classes of G correspond to partitions of n (A000041), this makes it relatively easy to find terms.
a(n) appears to equal n! * A082733(n).
(End)
REFERENCES
John F. Humphreys, A Course In Group Theory, Oxford Science Publications, 1996, chapter 10.
LINKS
Hiroaki Yamanouchi, Table of n, a(n) for n = 0..60
Eric Weisstein's World of Mathematics, Symmetric Group
CROSSREFS
KEYWORD
nonn
AUTHOR
Chad Brewbaker, Mar 27 2014
EXTENSIONS
a(8)-a(9) from Giovanni Resta, Mar 27 2014
More terms from Paul Boddington, Feb 23 2015
STATUS
approved