tailieunhanh - Báo cáo toán học: "Cycle index, weight enumerator, and Tutte polynomial"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Cycle index, weight enumerator, and Tutte polynomial. | Cycle index weight enumerator and Tutte polynomial Peter J. Cameron School of Mathematical Sciences Queen Mary University of London Mile End Road London E1 4NS . Submitted January 9 2002 Accepted February 27 2002 Abstract With every linear code is associated a permutation group whose cycle index is the weight enumerator of the code up to a trivial normalisation . There is a class of permutation groups the IBIS groups which includes the groups obtained from codes as above. With every IBIS group is associated a matroid in the case of a group from a code the matroid differs only trivially from that which arises directly from the code. In this case the Tutte polynomial of the code specialises to the weight enumerator by Greene s Theorem and hence also to the cycle index. However in another subclass of IBIS groups the base-transitive groups the Tutte polynomial can be derived from the cycle index but not vice versa. I propose a polynomial for IBIS groups which generalises both Tutte polynomial and cycle index. 1 Cycle index This note contains some remarks on the relations between the cycle index of a permutation group the weight enumerator of a linear code and the Tutte polynomial of a matroid. For more information on permutation groups codes and matroids see 6 10 14 respectively. Let G be a permutation group on a set Q where Q n. For each element g 2 G let ci g be the number of i-cycles occurring in the cycle decomposition of g. Now the cycle index of G is the polynomial Z G in indeterminates 51 . sn given by Z G G z s M- G g2G This can be regarded as a multivariate probability generating function for the cycle structure of a random element of G chosen from the uniform distribution . In particular PG x Z G s1 X si 1 for i 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 9 2002 N2 1 is the probability generating function for the number of fixed points of a random element of G so that substituting x 0 gives the proportion of derangements in G. Here and