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A139561
Growth function for the relatively free monoid on three generators with identity xyzyx = yxzxy.
0
1, 3, 9, 27, 78, 216, 568, 1416, 3327
OFFSET
0,2
COMMENTS
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.
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.
Shneerson has shown that this sequence grows roughly like some power of n^logn . Contrast to A140348, which has polynomial growth.
REFERENCES
L. M. Shneerson, Relatively free semigroups of intermediate growth, J. Algebra, 235 (2001) 484-546.
EXAMPLE
Substituting x->a, y->b, z->1 in the identity gives abba=baab, this is the smallest example.
Substituting x->ab, y->b, z->c in the identity gives abbcbab=babcabb.
CROSSREFS
Cf. A140348.
Sequence in context: A027027 A361845 A140348 * A152169 A370142 A241574
KEYWORD
nonn
AUTHOR
STATUS
approved