A239841 Ordered pairs of permutation functions on n elements where f(g(g(x))) = g(g(f(x))).

1, 1, 4, 30, 312, 3720, 64080, 1305360, 33949440, 1019692800, 36360576000, 1487539468800, 69633899596800, 3649476307276800, 213929162589542400, 13848506938506240000, 986705192227442688000, 76724136092268048384000, 6491471142159880740864000

Offset: 0

Author

Chad Brewbaker, Mar 27 2014

Keywords

• nonn

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

Cf. A000012, A000085, A000142, A088311, A053529, A001044.

Extensions

a(8)-a(9) from Giovanni Resta, Mar 27 2014

More terms from Paul Boddington, Feb 23 2015