Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Counting Lattice Paths by. | Counting Lattice Paths by Narayana Polynomials Robert A. Sulanke Boise State University Boise Idaho USA sulanke@ Submitted April 15 2000 Accepted August 3 2000 Abstract Let d n count the lattice paths from 0 0 to n n using the steps 0 1 1 0 and 1 1 . Let e n count the lattice paths from 0 0 to n n with permitted steps from the step set N X N 0 0 where N denotes the nonnegative integers. We give a bijective proof of the identity e n 2n-1 d n for n 1. In giving perspective for our proof we consider bijections between sets of lattice paths defined on various sets of permitted steps which yield path counts related to the Narayana polynomials. Mathematical Reviews Subject Classification 05A15 Key words Lattice paths Narayana numbers Delannoy numbers. 1 Introduction In the plane Z X Z let D n denote the set of all lattice paths from 0 0 to n n using steps from the step set 0 1 1 0 1 1 . The path counts D n n 0 1 3 13 63 321 1683 . are the well-known central Delannoy numbers. Let E n denote the set of all lattice paths from 0 0 to n n with permitted steps from the step set N X N 0 0 where N denotes the nonnegative integers. Figure 1a 1f resp. illustrates a path in E 8 D 8 resp. if the first step is ignored. We will give a bijective proof for the identity IE n 2n-1 D n for n 1 as requested in Stanley s 4 Exercise . A proof using generating functions appears in 4 . To give perspective for our proof we will consider bijections 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R40 2 between various sets of lattice paths which have path counts related to Narayana polynomials. The known results of the first two sections serve as background for the results of the last two sections. This paper will consider six step sets which are labeled arbitrarily by Si for i 1. 6. Table 1 contains a list of these step sets and an index to the path sets considered. Given a specific step set Si let Ai n denote the set of all lattice paths running from 0 1 to n n that use
