The Polya cycle index has been a key tool in understanding what a typical permutation π ∈ Sn ”looks like”. It is useful for studying properties of a permutation which depend only on its cycle structure. Here are some examples of theorems which can be proved using the cycle index. Lloyd and Shepp [25] showed that for any i < ∞, the joint distribution of (a1(π), · · · , ai(π)) for π chosen unifor...

Computing cycle index polynomial of a permutation group is known to be #P -complete. In this report we give some introduction to the problem and describe some permutation groups for which the computation of cycle index polynomial is easy.

Index 74 Preface The three subjects of the title (codes, matroids, and permutation groups) have many interconnections. In particular, in each case, there is a polynomial which captures a lot of information about the structure: we have the weight enumerator of a code, the Tutte polynomial (or rank polynomial) of a matroid, and the cycle index of a permutation group. With any code is associated a...

A k-partial permutation out of n elements is a k-tuple (p1, p2, . . . , pk) with k distinct elements and pi ∈ [n] = {1, 2, . . . , n}, i = 1, 2, . . . , k. Let (p1, p2 . . . , pn) be a full permutation of size n, where the elements are distinct and pi ∈ [n], i = 1, 2, . . . , n. Then we say the k-partial permutation (p1, p2, . . . , pk) is induced from the full permutation. A universal cycle on...