| Enumerative combinatorics is one of fundamental and important branches in Combinatorics.It mainly investigates counting problems of combinatorial settings under given conditions.Riordan arrays can deal with many problems in enumerative combinatorics,and it is an important tool to solve the counting problems.Riordan arrays are mainly used to characterize combinatorial problems and prove inequality relationships.In addition,we can also use them to prove combinatorial identity relationships,investigate inversion relationships,and enumerate various kinds of lattice paths.In recent years,there have been extensive studies on the properites of Riordan arrays and their applications,done by both national and international scholars.This paper mainly studies the application of Riordan arrays in enumerative combinatorics.First,we introduce the development of Riordan arrays and the theories of Riordan arrays.In addition,we present the basic definitions and properties of generating functions,Riordan arrays,the definitions of paths.In particular,we present three important paths:Dyck path,Motzkin path and Schroder path.Then we focus on two Bell-type Riordan arrays[r（n,k）]n,k≥0 and[s（n,k）]n,k≥0where r（n,k）and s（n,k）are the numbers of Schr（?）der paths and little Schr（?）der paths of length 2n with k hills,respectively.Using Riordan theory,we obtain four recurrence relations and also provide combinatorial proofs of these results.And we consider a new statistic,called initial ascending run and denoted iar,whose distribution on separable permutations is give by r（n,k）as well.That is p（n,k）and r（n-1,k-1）satisfy the same recurrence relations,where p（n,k）is the number of separable permutations of length n with iar=k.Based on the relations between separable permutations and di-sk trees,we present combinatorial proofs for the recurrence relations.Lastly,combining the path decomposition and the A-and Z-sequence of the Riordan array,we generalize the two main theorems in the paper and obtain two weighted recurrence relations.Based on the addition of parameters,we get quite a few Riordan arrays as special cases.And we can do a great deal of research on statistics iar.We present the refined results with iar of permutations avoiding one of the patterns among 123,132,213,231,312 and 321,and supply combinatorial proofs of these results. |
