Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: An Asymptotic Expansion for the Number of Permutations with a Certain Number of Inversions. | An Asymptotic Expansion for the Number of Permutations with a Certain Number of Inversions Lane Clark Department of Mathematics Southern Illinois University Carbondale Carbondale IL 62901-4408 USA lclark@ Submitted December 17 1998 Accepted August 8 2000 Abstract Let b n k denote the number of permutations of 1 . . ng with precisely k inversions. We represent b n k as a real trigonometric integral and then use the method of Laplace to give a complete asymptotic expansion of the integral. Among the consequences we have a complete asymptotic expansion for b n k n for a range of k including the maximum of the b n k n . AMS Subject Classification 05A16 05A15 05A10 A permutation ơ ơ 1 . ơ nỴ of n 1 . ng has an inversion at i j where 1 i j n if and only if ơ i ơ j . Let b n k denote the number of permutations of n with precisely k inversions. Then b n k b n k for all integers k while b n k 0 if and only if 0 k n. Bender 2 p. 110 showed that the b n k are log concave in k. Hence the maximum B n of the b n k occurs at k _ 2j as well as r 2 2 for odd 2 . See 3 pps. 236-240 for further results. Random permutations show see 3 pps. 282-283 for example that the b n k satisfy a central limit theorem with ịin 2 2 and Ơ2 n n 1 2n 5 72 see 2 Theorem 1 . Bender 2 p. 109 remarks that the theorems of Section 4 do not apply to the b n k . He then shows 2 p. 110 that the b n k are log concave in k so that Lemma 2 applies. This will give a first term asymptotic formula for b n k n when k _pn xơnJ where x is a fixed real number. In this paper we represent b n k as a real trigonometric integral. We then use the method of Laplace to give a complete asymptotic expansion of this integral in terms of the Bernoulli numbers and Hermite polynomials. Hence we have the complete asymptotic 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R50 2 expansion b n k n 6 2 1 2n 3 2e x2 2 2m 2 1 X q 1 -2 q S2q n H2q 2 1 2x i ln2m2 1 n O I I nm 3 2 1 as n 1 when 2k n xn3 2 3 where x2 x2 n Inn and
