ABSTRACT
This chapter covers a variety of topics illustrating different aspects of enumerative combinatorics and probability. The treatment of each topic is essentially self-contained.
This section illustrates another technique for enumerating certain collections of lattice paths. The basic idea is to introduce an equivalence relation on paths by cyclically shifting the steps of a path. A similar idea was used in §3.14 to enumerate lists of terms. 12.1. Theorem: Enumeration of Rational-Slope Dyck Paths. Let r and s be positive integers such that gcd(r, s) = 1. The number of lattice paths from (0, 0) to (r, s) that never go below the diagonal line sx = ry is
r + s
( r + s
r, s
) .
