[seqfan] Counting functions for groups

David Newman davidsnewman at gmail.com
Fri Oct 14 07:23:35 CEST 2011

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

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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