Some Properties Of Derangement Exponential Family And The Cycle Index Of Symmetry Group

This paper mainly focuses on two aspects:the one are some properties in derangement exponential family, and the other are some properties in the cycle index of symmetry group. Firstly,the card, deck, hand, exponential family and exponential formula are proposed by H.S.Wilf for the first time, which are mainly used to deal with the counting problems of connecting block structures. The derangement is a permutation with no fixed points, that the Dn denote n-derangement number. At present, the mainly research methods of derangement number are enumeration method and recursive method. Based on exponential family, this dissertation designs a new exponential family of the derangement and gets some properties. Secondly, the research of the cycle index of symmetry group has a very wide range. Given a non-negative integer sequence a={α1, α2, α3,…}, here α1+2α2+3α3+…=n, we consider how many replacements of n-permutation which has1-cycle with the long of α1,2-cycle with the long of α2,3-cycle with the long of α3, and so on. For a given permutationσ, the vector α=α(σ) represents circulation type of a. Thus we can get each length of cycle number of σ. Here c(a) represents the required number of permutation, Ψn (x) represents the cycle index of symmetry group. In this dissertation, we study the relationship between the Bell polynomial and the cycle index of symmetry group, and get some properties of c(a) and Ψ/n(x).The whole work in this paper can be summarized as follows:1.Overview the status research of exponential family and the cycle index of symmetry group, introducing the definitions and relating conclusions of exponential formula, Bell polynomial, the cycle index of symmetry group and binomial polynomial sequences; 2.Based on the concepts of exponential family, give the relationship and expression of card, deck and hand in derangement exponential family;3.Study some properties of the derangement exponential family, the relationship between two classes of Stirling number is given, getting some new identities about derangement number;4.Obtain the relationship between the cycle index of symmetry group and the exponential Bell polynomial as well as the complete Bell polynomial, By Ramanujan’s formula and some properties of the divisor function, getting the equation with the cycle index of symmetry group and the prime not equal to five;5.Research the application of the cycle index of symmetry group in the first kind of Stirling number and the binomial polynomial sequence and convolution of polynomial sequences.