Conjugacy Classes in Order- Preserving Transformation Semi groups with Injective Contraction

Enumerating the elements within transformation semigroups poses a significant challenge. Prior knowledge has been more on the injective order-preserving and order-decreasing transformation semigroup, a sub-semigroup of the injective transformation semigroup. This work categorized elements within the injective order-preserving sub-semigroup with contraction, arranging them into conjugacy classes using a path decomposition approach based on circuit and proper paths. Furthermore, these conjugacy classes were organized according to the number of images. A general expression was derived for the number of conjugacy classes in the injective order-preserving contraction transformation semigroup. We found that the number of conjugacy classes in this transformation is precisely given by the sequence A000070 (OEIS). This alignment suggests a profound relationship between the structure of these transformations and the theory of partitions (number theory), opening up new avenues for research and analysis.