Enumeration of finite dynamical systems with given number of leaves

The number of leaves in a dynamical system’s functional graph is a crucial combinatorial invariant, directly encoding the cardinality of the system’s image and reflecting its non-injectivity. This paper presents an enumeration of nonequivalent finite dynamical systems with a prescribed number of leaves. We develop a comprehensive combinatorial framework that counts both transitive and general systems with given cycle length m, height k, and exactly l leaves. By integrating the graph-theoretic structure of functional graphs with advanced enumerative combinatorics, including Möbius inversion, and symmetry analysis via cyclic group actions, we derive explicit formulas for the quantities fk,m(n,l) and its labeled counterpart f¯k,m(n,l), where fk,m(n,l) denotes the number of unlabeled rooted-tree cycles of order n with a cycle of order m≥1, maximal height k, and exactly l leaves, while f¯k,m(n,l) counts the corresponding labeled configurations.