Some problems of the theory of enumerating the permutations with restricted positions

We sketch the circle of fundamental ideas and trends in the development of one of the important branches of combinatorial analysis—the theory of enumerating the permutations with restricted positions, which dates back to Euler, Monmort, Lucas, Cayley, and other classical authors. We devote a great deal of attention to new and up-to-date trends of the theory, some of which are still in the formative stage. We state 10 conjectures and problems.