“…If we require each block/cycle/ordered block (not only the non-distinguished ones) to contain at least s elements, then we call them n, m, r , D A ≥s, * n, m, r , DL ≥ s, * n, m, r , respectively. Alternative s-associated r -Bell numbers appear in two recent papers [1,3] (the former one allows a more general restriction on the cardinality of the blocks), while in [19,24], the authors investigated alternative 2-associated and alternative s-associated r -factorials under the names of r -derangement and generalized r -derangement numbers.…”