The cardinality of Kiselman's semigroups grows double-exponentially

Abstract

Let $K_n$ denote Ganyushkin-Kudryavtseva-Mazorchuk's generalization of Kiselman's semigroups. We show that the sequence $2^{-n/2}\cdot \log|K_n|$ admits finite limits as $n$ grows to infinity both on odd and even values.