Rooted cyclic permutations of lattice paths and uniform partitions

A partition of a given set is said to be uniform if all the partition classes have the same cardinality. In this paper, we will introduce the concepts of rooted nn-lattice path and rooted cyclic permutation and prove some fundamental theorems concerning the actions of rooted cyclic permutations on rooted lattice nn-paths. The main results obtained have important applications in finding new uniform partitions. Many uniform partitions of combinatorial structures are special cases or consequences of our main theorems.