Combinatorial And Algebraic Properties Of Nilpotent And Idempotent Conjugacy Classes: A Study In Partial One-To-One Transformation Semigroups

This paper investigates the combinatorial and algebraic properties of nilpotent and idempotent conjugacy classes in partial one-to-one transformation semigroups ????????. By analyzing the total number of conjugacy classes and the cardinalities of path (chain) decompositions, we establish explicit formulas and sequences that highlight the intricate relationships within these structures. Specifically, we derive the expressions for ????????, representing the total number of nilpotent conjugacy classes, ????????, the total number of idempotent conjugacy classes, and also ???????? and ????????, that captures the cardinality of chains in the chain decomposition of nilpotent and idempotent conjugacy classes and we also present a detailed table showcasing these values of sequences for different ????. The obtained results provide a deeper understanding of the interplay between combinatorial and algebraic aspects in the context of transformation semigroups, offering a solid foundation for further investigations in this area of mathematics.