  A square matrix is called {\it Hessenberg} whenever each entry below the
subdiagonal is zero and each entry on the subdiagonal is nonzero. Let $V$
denote a nonzero finite-dimensional vector space over a field $\fld$. We
consider an ordered pair of linear transformations $A: V \rightarrow V$ and
$A^*: V \rightarrow V$ which satisfy both (i), (ii) below. (i) There exists a
basis for $V$ with respect to which the matrix representing $A$ is Hessenberg
and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for
$V$ with respect to which the matrix representing $A$ is diagonal and the
matrix representing $A^*$ is Hessenberg.
  \noindent We call such a pair a {\it thin Hessenberg pair} (or {\it TH
pair}). By the {\it diameter} of the pair we mean the dimension of $V$ minus
one. There is an "oriented" version of a TH pair called a TH system. In this
paper we investigate a connection between TH systems and double Vandermonde
matrices. We give a bijection between any two of the following three sets:
  \cdot The set of isomorphism classes of TH systems over $\K$ of diameter $d$.
  \cdot The set of normalized west-south Vandermonde systems in $\Mdf$.
  \cdot The set of parameter arrays over $\K$ of diameter $d$. We give a
bijection between any two of the following five sets:
  \cdot The set of affine isomorphism classes of TH systems over $\K$ of
diameter $d$.
  \cdot The set of isomorphism classes of RTH systems over $\K$ of diameter
$d$.
  \cdot The set of affine classes of normalized west-south Vandermonde systems
in $\Mdf$.
  \cdot The set of normalized west-south Vandermonde matrices in $\Mdf$.
  \cdot The set of reduced parameter arrays over $\K$ of diameter $d$.
