Bicoloured Dyck paths and the contact polynomial for n non-intersecting paths in a half-plane lattice -: art. no. R35

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) over cap (H)(2r)(n; kappa) = Sigma(c=1)(r+1)\H-2r((n)) (c)\kappa(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 kappa by the length generating function for Dyck paths, kappa(omega) = Sigma(r=0)(infinity) 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 (Z) over cap (H)(24) (1; kappa(omega)) = Sigma(b=0)(infinity) C-r+b!omega(b). 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 colour can only change at a contact with the line y = 0. For n > 1, the coefficient of omega(b) in (Z) over cap (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.