Ballot Permutations And Related Enumeration Problems

Ballot permutations are permutations with restriction on their descents and ascents set.Enumeration on restricted permutations is an important research direction in enumerative combinatorics.In this paper,we study Ballot permutations and related enumeration problems,and get a combinatorial proof of the recurrence relations of Ballot permutations.In the second chapter,we define quasi-Dyck permutations,then,we construct the splitting and merging methods of permutations.Next,we obtain properties of quasiDyck permutations and Ballot permutations.Finally,we construct a bijection of Ballot permutations of length 2n and quasi-Dyck permutations of length 2n-1.Using this bijection,we get a combinatorial proof of the recurrence relation of Ballot permutations.In the third chapter,we get a combinatorial proof of the recurrence relation of odd order permutations and Callan permutations.