Enumeration On Permutations With Restrictions On Cycle Length Or Left-to-Right Minima

Permutations are basic research objects in enumeration combinatorics.Enumeration on restricted permutations and related statistics is an important research topic.Previous research has shown that the number of Callan permutations is the same as odd order permutations.In this paper,we give a combinatorial proof of this conclusion.Then we define k-Callan permutations and give the enumeration.And for any integer k>1,we give the bijection between k-Callan permutations and permutations whose cycle lengths aren't divisble by k.Moreover,we enumerate k-Callan permutations according to the number of inversions and give the corresponding q-polynomial.