Combinatorial Sequences And The Related Recurrence Relations

The topic of combinatorial sequence and its recurrence relation has been one of the core problems in combinatorics. In this thesis, we study the combinatorial sequence and its recurrence relation. We discuss four combinatorial sequences, including the binomial coefficients, the Stirling numbers of the two kinds and the Lah numbers. The analytic expressions of these combinatorial sequences are given(except for the Stirling numbers of the first kind). Moreover, we give the combinatorial interpretations and the recurrence relations of these sequences, and some related well-known combinatorial identities are also presented.In this thesis, we give the general solutions of the linear homogeneous and non-homogeneous recurrence relations with constant coefficients. On the one hand, by using symmetric function, we give a combinatorial proof of a non-linear recurrence relation of order 2.