Number of rotation subgroups of S_n

[Ross La Haye]

This may be a stretch, but I've been thinking about a particular formula for
the number of k-length walks=trails=paths in the Boolean algebra of order n:
(2n)!! / ((n-k)!2^k) (applies to A090802).  (2n)!! is the order of the
automorphism group for the n-cube (A000165) and I was wondering what its
subgroups were like, mainly due to the denominator in the formula...In other
words, it seems like the denominator is partitioning the elements of the
automorphism group in some way, but maybe I'm confused...At any rate, if I'm
not mistaken, the automorphism group of the n-cube is constructed by
rotations and reflections...so perhaps there may be some intersection
between my interest and the number of rotation subgroups of S_n...probably a
stretch, like I said, but thought I'd throw it out just in case...BTW, isn't
S_n itself an automorphism group...?

Ross



-----Original Message-----
From: N. J. A. Sloane [mailto:njas at research.att.com]
Sent: Monday, October 10, 2005 1:19 PM
To: seqfan at ext.jussieu.fr
Cc: njas at research.att.com
Subject: Number of rotation subgroups of S_n



A friend suggested the following set of sequences

Let S_n be the symmetric group of all perms of n letters,
of order n!

A transposition is a permutation like (3,5).
A reflection is any element of order 2, that is, a product
   of disjoint transpositions
A "rotation" is a product of two transpositions, not necessarily disjoint,
   so either (1,2,3) or (1,2)(3,4), etc.

Questions:
How many subgroups of S_n have the property that they are
generated by rotations?
How many non-isomorphic ones?

How many subgroups of S_n have the property that they are
generated by reflections?  (Is this the Bell numbers, A000110 ?)
How many non-isomorphic ones?  (Again, this may be a well-known sequence)

Same two questions for groups generated by rotations AND reflections?

NJAS