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{\bf R. Brak and  J. W. Essam}
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{\bf Bicoloured Dyck Paths and the Contact Polynomial for $n$
Non-Intersecting Paths in a Half-Plane Lattice}
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In this paper configurations of $n$ non-intersecting lattice paths
which begin and end on the line $y=0$ and are excluded from the region
below this line are considered. Such configurations are called Hankel
$n-$paths and their contact polynomial is defined by
$\hat{Z}^{\cal{H}}_{2r}(n;\kappa)\equiv \sum_{c= 1}^{r+1} |{\cal
H}_{2r}^{(n)}(c)|\kappa^c$ where ${\cal H}_{2r}^{(n)}(c)$ is the set
of Hankel $n$-paths which make $c$ intersections with the line $y=0$
the lowest of which has length $2r$. These configurations may also be
described as parallel Dyck paths.

It is found that replacing $\kappa$ by the length generating function
for Dyck paths, $\kappa(\omega) \equiv \sum_{r=0}^\infty C_r
\omega^r$, where $C_r$ is the $r^{th}$ Catalan number, results in a
remarkable simplification of the coefficients of the contact
polynomial. In particular it is shown that the polynomial for
configurations of a single Dyck path has the expansion
$\hat{Z}^{\cal{H}}_{2r}(1;\kappa(\omega)) = \sum_{b=0}^\infty
C_{r+b}\omega^b$.  This result is derived using a bijection between
bi-coloured Dyck paths and plain Dyck paths.  A bi-coloured Dyck path
is a Dyck path in which each edge is coloured either red or blue with
the constraint that the colour can only change at a contact with the
line $y=0$.  For $n>1$, the coefficient of $\omega^b$ in
$\hat{Z}^{\cal{W}}_{2r}(n;\kappa(\omega))$ is expressed as a
determinant of Catalan numbers which has a combinatorial
interpretation in terms of a modified class of $n$ non-intersecting
Dyck paths. The determinant satisfies a recurrence relation which
leads to the proof of a product form for the coefficients in the
$\omega$ expansion of the contact polynomial.

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