Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: On counting permutations by pairs of congruence classes of major index. | On counting permutations by pairs of congruence classes of major index Helene Barcelo and Robert Maule Department of Mathematics Arizona State University Tempe AZ barcelo@ rgmaule@ Sheila Sundaram 240 Franklin St. Ext. Danbury CT 06811 sheila@ Submitted August 16 2001 Accepted March 19 2002. MR Subject Classification Primary 05E25 Abstract For a fixed positive integer n let Sn denote the symmetric group of n permutations on n symbols and let maj ơ denote the major index of a permutation Ơ. Fix positive integers k n and nonnegative integers i j. Let mn i k j denote the cardinality of the set ơ G Sn maj ơ i mod k maj ơ-1 j mod . In this paper we use combinatorial methods to investigate these numbers. Results of Gordon and Roselle imply that when k are relatively prime n mn i k j k-f. We give a combinatorial proof of this in the case when divides n 1 and k divides n. 1 Introduction Denote by Sn the symmetric group of all n permutations on the n symbols 1 . n. First recall some combinatorial definitions pertaining to permutations. See . St . Definition Let ơ G Sn. For 1 i n 1 i is said to be a descent of Ơ if ơ í ơ i 1 . Definition The major index of Ơ denoted maj a is the sum of the descents of Ơ. The values of the statistic maj range from 0 for the identity to n . In BS the following result was discovered using certain representations of the symmetric group Sn and then proved by means of a bijection as well. Proposition BS Theorem Fix an integer 0 i n 1. Then n 1 a G Sn maj a i mod n . This paper is similarly motivated by the following algebraic discovery. By considering characters of the symmetric group that are induced from irreducible characters of the cyclic subgroup of order n and computing intertwining numbers the following identity was discovered for fixed nonnegative integers i and j Su . n 2 G Sn maj a i mod n maj ơ-1 j mod n 1 A the electronic journal of combinatorics 9 2002 R21 1 Let mn i k j l be .
