Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Bicoloured Dyck paths and the contact polynomial for n non-intersecting paths in a half-plane lattice. | Bicoloured Dyck paths and the contact polynomial for n non-intersecting paths in a half-plane lattice. R. Brakf and J. W. EssamỊ fDepartment of Mathematics The University of Melbourne Parkville Victoria 3052 Australia Department of Mathematics Royal Holloway College University of London Egham Surrey TW20 0EX England. Submitted May 14 2001 Accepted Feb 22 2003 Published Sep 12 2003 MR Subject Classifications 05A15 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 and their contact polynomial is defined bv ZUn K. r 1 7- n I Kc where 7- d is rhe set of Unnkel Ii-iuillis uenneu y 2r n K c1 I I 2r C IK wiieie I 2r C is bile sei oi j-j-aiiiA-ei n paIllis 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 K by the length generating function for Dyck paths k w EAo Crwr where Cr 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 z2r 1 k w Vb_0 Cr bwb. 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 wb in Zfr n k w 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 w expansion of the contact polynomial. email THE .
