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Counting Strings with Given Elementary Symmetric Function
Evaluations II: Circular Strings

*Frank Ruskey*,
Department of Computer Science,
University of Victoria, Canada.

*C. Robert Miers*,
Department of Mathematics and Statistics,
University of Victoria, Canada.

### Abstract:

Let **a** be a string over an alphabet that is a finite ring, *R*.
The *k*-th elementary symmetric function evaluated at **a** is
denoted *T*_{k}(**a**).
In a companion paper we studied the properties of
**S**_{R} ( *n*;
*t*_{1},*t*_{2},...,*t*_{k} ),
the set of of length *n* strings for which
*T*_{i} ( **a** ) = *t*_{i}.
Here we consider the set,
**L**_{R} ( *n*;
*t*_{1},*t*_{2},...,*t*_{k} ),
of equivalence classes under rotation
of aperiodic strings in
**S**_{R} ( *n*;
*t*_{1},*t*_{2},...,*t*_{k} ),
sometimes called Lyndon words.
General formulae are established, and then refined for the cases
where *R* is the ring of integers **Z**_{q}
or the finite field **Z**_{q}.

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