A139561 Growth function for the relatively free monoid on three generators with identity xyzyx = yxzxy.

1, 3, 9, 27, 78, 216, 568, 1416, 3327

Offset: 0

Author

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

Keywords

• nonn

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.

Examples

			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.
		

References

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

Crossrefs

Cf. A140348.