A study of Eulerian numbers by means of an operator on permutations

AbstractIn a previous paper an operator on permutations was defined and its application was discussed. The operator preserves the numbers of their ascents, and each permutation has its own period and orbit under the operator, by which it enables us to study Eulerian numbers. The objective of this paper is to investigate the numbers of orbits and permutations in more detail and to discuss their applications to Eulerian numbers. They include representations of Eulerian numbers by means of orbits and some congruence relations modulo prime powers