Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A Relationship Between the Major Index For Tableaux and the Charge Statistic For Permutations. | A Relationship Between the Major Index For Tableaux and the Charge Statistic For Permutations Kendra Killpatrick Pepperdine University Malibu California Submitted Jul 13 2005 Accepted Aug 30 2005 Published Sep 5 2005 Mathematics Subject Classifications 05A15 05E10 Abstract The widely studied -polynomial f x q which specializes when q 1 to fx the number of standard Young tableaux of shape A has multiple combinatorial interpretations. It represents the dimension of the unipotent representation Sq of the finite general linear group GLn q it occurs as a special case of the Kostka-Foulkes polynomials and it gives the generating function for the major index statistic on standard Young tableaux. Similarly the q-polynomial gx q has combinatorial interpretations as the q-multinomial coefficient as the dimension of the permutation representation Mx of the general linear group GLn q and as the generating function for both the inversion statistic and the charge statistic on permutations in Wq. It is a well known result that for A a partition of n dim MqX Y K dim Sg where the sum is over all partitions g of n and where the Kostka number K x gives the number of semistandard Young tableaux of shape g and content A. Thus gx q fx q is a q-polynomial with nonnegative coefficients. This paper gives a combinatorial proof of this result by defining an injection f from the set of standard Young tableaux of shape A SYT A to Wx such that maj T ch f T for T 2 SYT A . Key words Young tableaux permutation statistics inversion statistic charge statistic Kostka polynomials. 1 Introduction For A any partition of n fx gives the number of standard Young tableaux of shape A. The q-version of fx is a polynomial that has many important combinatorial interpretations. In particular f x q is known to give the dimension of the unipotent representation Sq THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R45 1 of the finite general linear group GLn q . The polynomial f x q
