# [seqfan] Re: Counting functions for groups

franktaw at netscape.net franktaw at netscape.net
Fri Oct 14 07:37:54 CEST 2011

```This is not a direct response to David's question. One of the things I
try to do when I see a proposed sequence is to try to find simpler
and/or more basic sequences based on the same idea(s).

In this case, combinatorially, a group with distinguished generators is
a different animal from just the group. Do we have a sequence with the
number of groups-with-generators with n elements? How about a table of
the number of groups with n elements and k generators?

In both of these cases, there are actually two sequences: one where the
set of generators is required to be minimal, and one where it is not
(so T(n,n) would be A000001(n)).

I do think all four of the sequences I described here do belong in the
OEIS. I suspect that others here are better placed than I am to compute
them. (I can't readily even compute enough of them to search and see if

-----Original Message-----
From: David Newman <davidsnewman at gmail.com>

I'd like to know if there is room in the OEIS for sequences of the
following
sort. (I've formulated the idea for 2 generators just to give the idea.
The
same can be formulated for any number of generators.)

Let G be a group generated by two generators.  For each element in the
group
there is a shortest way to write it as a product of these generators.
For a
given element g belonging to G, we'll say that the length of g is the
length
of this shortest product.  Form a sequence of the number of elements of
G
having length i, i=0,1,2,...  (Using the convention that the length of
the
identity is 0)  This sequence I'll call the counting function for the
group
G.

For example if the two generators are the permutations (1,2,4,5,3) and
(2,3,1,4,5) and the operation is composition of permutations, then the
sequence is 1,2,4,5,10,15,16,5,1 (not in the encyclopedia).  Different
groups may have the same counting function and one group may have
several
counting functions depending on which generators are chosen, but there
are
only a finite number of counting functions for all groups with a given
number of elements.

Is the OEIS the right place for these counting functions?  If not, is
there
a good place for them?

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```