Uniform Partitions of Lattice Paths and Chung-Feller Generalizations

The Chung-Feller Theorem shows that n + 1 divides (2n ) by considering all (2n) lattice paths consisting of n upsteps and n downsteps and uniformly partitioning this set into n + 1 equivalence classes. (A set is uniformly partitioned if all partition classes have the same cardinality.) This partitioning is done by exhibiting a parameter (such as the number of upsteps above ground) that is uniformly distributed over the values 0 through n [2], [3, pp. 65-77]. In this paper we present a new uniformly distributed parameter based on the rightmost lowest point of the path that provides a quick proof of the Chung-Feller Theorem and has immediate generalizations. We consider those lattice paths in the Cartesian plane beginning at (0, 0) and proceeding in diagonal step U = (1, 1) (an upstep) or D = (1, -1) (a downstep). There is a one-to-one correspondence between lattice paths and finite strings from the alphabet {U, D}. Let A(n, k) be the set of all lattice paths ending at the point (n, k) (the terminal point). For example, the path P = UDUDDUUDUDDDUUUU e B(16, 2) is illustrated in Figure 1.