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Báo cáo toán học: "Two Remarks on Skew Tableaux"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Two Remarks on Skew Tableaux. | Two Remarks on Skew Tableaux Richard P. Stanley Department of Mathematics Massachusetts Institute of Technology Cambridge MA 02139 rstan@math.mit.edu Submitted Feb 22 2011 Accepted Jun 27 2011 Published Jul 15 2011 Mathematics Subject Classification 05E05 Abstract This paper contains two results on the number f ơ t of standard skew Young tableaux of shape ơ t. The first concerns generating functions for certain classes of periodic shapes related to work of Gessel-Viennot and Baryshnikov-Romik. The second result gives an evaluation of the skew Schur function A M x at x 1 1 22k 1 32k . for k 1 2 3 in terms of f ơ t for a certain skew shape ơ t depending on x p. 1 Introduction We assume familiarity with the basic theory of symmetric functions and tableaux from 6 Chap. 7 . Baryshnikov and Romik 1 obtain explicit formulas for the number of standard Young tableaux of certain skew shapes which they call diagonal strips. As the strips become thicker their formulas becomes more and more complicated. In the next section we give simple generating functions for certain classes of diagonal strips of arbitrarily large thickness. Our results are neither a subset nor superset of those of Baryshnikov and Romik. Our results in Section 2 were also obtained in unpublished work of Gessel and Viennot 3 11 but our approach is more straightforward. In Section 3 we obtain a formula for the evaluation of skew Schur functions A M x at x 1 1 22k 1 32k . for k 1 2 3. These formulas have the following form. Let n A p the number of squares in the diagram of A p and choose m Al. Then A M 1 I 22k 1 32k . c n This author s contribution is based upon work supported by the National Science Foundation under Grant No. 0604423. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2 2011 p16 1 where ơ r is a certain skew shape depending on A p and m. Moreover f ơ t denotes the number of standard Young tableaux of shape ơ r and c n m is a simple explicit function. The special case ơ r 1 the unique partition of 1 .