# [seqfan] Re: Counting functions for groups

Roland Bacher Roland.Bacher at ujf-grenoble.fr
Fri Oct 14 09:15:03 CEST 2011

```I believe that such length-counting functions are still more
important for infinite groups.

There are a few important groups which have more or less canonical
generators: Coxeter groups and especially Weil groups, braid groups,
free (abelian or non-abelian) groups, free nilpotent groups with
a given
number of generators and of given deepth, Thompson's group, the
Grigorchuk group etc. Most of their length functions are probably
already in the OEIS since they appear in the litterature.

Roland Bacher

On Fri, Oct 14, 2011 at 01:37:54AM -0400, franktaw at netscape.net wrote:
> 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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