Enumeration Of Crossings On Non-nesting Matching

In linear representation of non-nesting matching, junctions of arcs are called crossings. The major purpose of this article is to count the crossings in non-nesting matchings. One of counting methods of enumerative combinatorics is to establish bijection between one combinatorial object and another combinatorial object. Conclusion can be obtained indirectly by existing methodology of counting of enumerative combinatorics. This article adopted the theory of bijection to establish the one between non-nesting matching related to crossings of statistics and Dyck path related to the area size of statistics. Then the relation between non-nesting matching and Dyck path will be made by the inverse method. The enumeration of Dyck Paths is Catalan number, a recurrence formula can be acquired of non-nesting matching with the length of 2n and k crossings by computing method of recurrence formula of Dyck path. Further more, a generating function is obtained. At the same time, we explored the way of partition Catalan number based on crossings. Then, enumeration formula of crossing formally will be defined based on q-Catalan polynomial.