[seqfan] No. of words of length n in certain group

N. J. A. Sloane njas at research.att.com
Sun Nov 22 04:11:09 CET 2009

To Math Fun, Seq Fan:

I ran into my distinguished (former) colleague Colin Mallows yesterday,
and he said that it would be nice if someone would extend his sequence A154638.

Take the infinite reflection group with 5 generators S_1, ..., S_5,
satisfying (S_i)^2 = (S_i S_j)^3 = I,
and let a(n) be number of distinct elements whose minimal representation
as a product of generators has length n:

1, 5, 20, 70, 240, 780, 2730, .. (for n>= 0)

Can anybody help extend this sequence?  Is there a generating function?
What about other groups?  This might be a gaping hole in the OEIS!
There must be a huge literature on this problem.

The books that I know about that might be related,
Lyndom & Schupp, Combinatorial Group Theory,
Magnus, Karrass, Solitar, same title,
Johnson, Presentations of Groups,
aren't exactly full of sequences, as far as I can tell.

Can some expert please help?


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