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@AnthonyLabarre Yes. I believe that these permutations of length 9 were the shortest permutations that I found which give non-unimodal growth. However, I don't think that I did an exhaustive search. I think I took arbitrary permutations and then calculated the counting function for these pairs.
@AnthonyLabarre I couldn't wait and did the calculations by hand. For words of length 4, (which is the first case where we disagree) using "a" for one permutation and "b" for the other, I find that aaaa = bbbb = the identity, while aabb = bbaa. Then, by my count, there are 16-3 = 13 words of length 4. Is this of some help?
@Anthony Labarre I am away from my computer for the next few days. Hopefully when I get back I'll be able to find my computations in my Mathematica files. If I have to redo the computations it will take me a substantially longer time.
