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 Z^H _2r(n;κ)_c= 1^r+1 |H_2r^(n)(c)|κ^c where 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 κ by the length generating function for Dyck paths, κ(ω) _r=0^∞ C_r ω^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 Z^H _2r(1;κ(ω)) = _b=0^∞ C_r+bω^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 ω^b in Z^W _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. , number=1, journal=The Electronic Journal of Combinatorics, author=Brak, R. and Essam, J. W., year=2003, month=Sep., pages=#R35