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: averages: Remarks on a paper by Stanley. | Plancherel averages Remarks on a paper by Stanley Grigori Olshanski Institute for Information Transmission Problems Bolshoy Karetny 19 Moscow 127994 GSP-4 Russia and Independent University of Moscow Russia olsh2007@ Submitted Oct 1 2009 Accepted Mar 10 2010 Published Mar 15 2010 Mathematics Subject Classification 05E05 Abstract Let Mn stand for the Plancherel measure on Yn the set of Young diagrams with n boxes. A recent result of R. P. Stanley arXiv says that for certain functions G defined on the set Y of all Young diagrams the average of G with respect to Mn depends on n polynomially. We propose two other proofs of this result together with a generalization to the Jack deformation of the Plancherel measure. 1 Introduction Let Y denote the set of all integer partitions which we identify with Young diagrams. For A G Y denote by A the number of boxes in A and by dim A the number of standard tableaux of shape A. Let also c1 A . C A A be the contents of the boxes of A written in an arbitrary order recall that the content of a box is the difference j i between its column number j and row number i . For each n 1 2 . denote by Yn c Y the finite set of diagrams with n boxes. The well-known Plancherel measure on Yn assigns weight dim A 2 n to a diagram A G Yn. This is a probability measure. Given a function F on the set Y of all Young diagrams let us define the nth Plancherel average of F as F n Ẹ M F A . AeYn Supported by a grant from the Utrecht University by the RFBR grant 08-01-00110 and by the project SFB 701 Bielefeld University . THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R43 1 In the recent paper 17 R. P. Stanley proves among other things the following result 17 Theorem Theorem . Let xi x2 . be an arbitrary symmetric function and set Gv A C1 A . C A A 0 0 . A G Y. Then G n is a polynomial function in n. The aim of the present note is to propose two other proofs of this result and a generalization which is related to the Jack
