Generalizations of Stirling-like and Bell-like Numbers

Stirling and Bell numbers play a fundamental role in enumerative combinatorics, they count, among others, partitions of a finite set. We give a survey on results concerning certain types of generalizations of these numbers. Throughout the paper, we assume that some distinguished elements are not allowed to be in the same block. We can define several variants: when we have restrictions on the cardinality of the blocks; colour certain elements under some colouring rules; or ordered modifications where the partition or the blocks are ordered. We collect the properties of these generalized Stirling-like and Bell-like numbers together with their combinatorial background. These include some new identities as well as proofs.