tailieunhanh - Báo cáo toán học: " On a Balanced Property of Derangements"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On a Balanced Property of Derangements. | On a Balanced Property of Derangements Miklós Bona Submitted Jun 6 2006 Accepted Oct 30 2006 Published Nov 6 2006 Mathematics Subject Classification 05A16 Abstract We prove an interesting fact describing the location of the roots of the generating polynomials of the numbers of derangements of length n counted by their number of cycles. We then use this result to prove that if k is the number of cycles of a randomly selected derangement of length n then the probability that k is congruent to a given r modulo a given q converges to 1 q. Finally we generalize our results to a-derangements which are permutations in which each cycle is longer than a. 1 Introduction Let c n k denote the number of permutations of length n with k cycles. The numbers c n k are then called the signless Stirling numbers of the first kind. It is well-known 3 that n Cn x c n k xk x x 1 x n 1 . 1 k 1 Setting x 1 this shows that there are as many permutations of length n with an odd number of cycles as there are with an even number of cycles. If instead of considering the sum of all Stirling numbers c n k so that n is fixed and k belongs to a certain residue class modulo 2 we consider the sum of the Stirling numbers c n k so that n is fixed and k belongs to a certain residue class modulo q the result is a little bit less compact. These sums will no longer be equal to n q but it will be true that as n goes to infinity the limit of any of these q sums divided by n will converge to 1 q. We will prove this fact in this paper as a way to illustrate our techniques. A derangement is a permutation with no fixed points cycles of length 1 . It is well-v n l 1 known 2 that number of derangements of length n is D n n i ol which is the University of Florida Gainesville FL 32611-8105. Partially supported by an NSA Young Investigator Award. Email bona@. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R102 1 integer closest to n e. Let d n k be the number of derangements of length n with k cycles. .