Bicoloured Dyck Paths and the Contact Polynomial for n Non-Intersecting Paths in a Half-Plane Lattice

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 ẐH 2r(n;κ) ≡ ∑r+1 c=1 |H (n) 2r (c)|κc where H (n) 2r (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 κ by the length generating function for Dyck paths, κ(ω) ≡ ∑∞r=0 Crω, where Cr is the r 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 ẐH 2r(1;κ(ω)) = ∑∞ b=0 Cr+bω . This result is derived using a bijection between bicoloured 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 email: [email protected], [email protected] the electronic journal of combinatorics 8 (2001), #R00 colour can only change at a contact with the line y = 0. For n > 1, the coefficient of ω in ẐW 2r (n;κ(ω)) 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 ω expansion of the contact polynomial. the electronic journal of combinatorics 8 (2001), #R00 i