Bicoloured Dyck paths and the contact polynomial for n non-intersecting paths in a half plane.

Abstract

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 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 by the length generating function for Dyck paths,
(!) P1
r=0 Cr!r, where C_r is the rth 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
^ ZH
2r(1; (!)) = P1
b=0 Cr+b!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 !b in ^ ZW
2r (n; (!)) 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 cofficients in the ! expansion of the contact polynomial.