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%I #5 Jan 31 2018 18:22:57
%S 1,3,9,27,78,216,568,1416,3327
%N Growth function for the relatively free monoid on three generators with identity xyzyx = yxzxy.
%C A semigroup which satisfies the identity xyzyx=yxzxy is called nilpotent of class 2 in the Malcev sense. Initially this sequence looks very like A140348, which counts the words that are distinct in the free nil-2 group.
%C The sequences first differ at n=7, where there are six equations that hold for the group but do not follow from the Malcev identity, e.g. abccaab = caabbca. Cancellation is not assumed and does not hold, so despite the fact that the former equation does not follow from the Malcev identity, aabccaab = acaabbca does follow.
%C Shneerson has shown that this sequence grows roughly like some power of n^logn . Contrast to A140348, which has polynomial growth.
%D L. M. Shneerson, Relatively free semigroups of intermediate growth, J. Algebra, 235 (2001) 484-546.
%e Substituting x->a, y->b, z->1 in the identity gives abba=baab, this is the smallest example.
%e Substituting x->ab, y->b, z->c in the identity gives abbcbab=babcabb.
%Y Cf. A140348.
%K nonn
%O 0,2
%A _David S. Newman_ and _Moshe Shmuel Newman_, Jun 10 2008
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