Rooted cyclic permutations of lattice paths and uniform partitions

Abstract

a b s t r a c tA 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 n-lattice path and rooted cyclic permutation and prove some fundamental theorems concerning the actions of rooted cyclic permutations on rooted lattice n-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.