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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 (list; graph; refs; listen; history; text; internal format)



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.


L. M. Shneerson, Relatively free semigroups of intermediate growth, J. Algebra, 235 (2001) 484-546.


Table of n, a(n) for n=0..8.


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.


Cf. A140348.

Sequence in context: A269488 A027027 A140348 * A152169 A241574 A268013

Adjacent sequences:  A139558 A139559 A139560 * A139562 A139563 A139564




David S. Newman and Moshe Shmuel Newman, Jun 10 2008



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Last modified February 18 09:39 EST 2020. Contains 332011 sequences. (Running on oeis4.)