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Báo cáo toán học: "On Descents in Standard Young Tableaux"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: On Descents in Standard Young Tableaux. | On Descents in Standard Young Tableaux Peter A. Hasto Department of Mathematics University of Helsinki P.O. Box 4 00014 Helsinki Finland peter.hasto@helsinki.fi. Submitted July 9 2000 Accepted December 4 2000 Abstract In this paper explicit formulae for the expectation and the variance of descent functions on random standard Young tableaux are presented. Using these it is shown that the normalized variance V E2 is bounded if and only if a certain inequality relating the tableau shape to the descent function holds. 1. Introduction In a recent paper Adin and Roichman defined and studied certain descent functions on standard Young tableaux see AR . They calculated the expectation value and derived an estimate for the variance of these functions. Their results were proved using character theory of symmetric groups. In this paper the expectation value and the variance are calculated using an elementary method. This method is based on the hook-bijection of Novelli Pak and Stoyanovskii cf. NPS . The expressions for the expectation and the variance are used to derive a somewhat more precise form of the results in AR . In the following section the necessary definitions are given. Thereafter the main results of the paper are stated. In the third section two auxiliary lemmata are presented. These lead directly to the proofs of the main results and some corollaries in the fourth section. Supported in part by the Austrian Academic Exchange Service Osterreichischer akademi-scher Austauschdienst . I d like to thank C. Krattenthaler for suggesting this topic to me. Mathematics Subject Classification 1991 primary 05E10 secondary 05A15 1 The electronic .journal of combinatorics 7 2000 R59 2 2. Some definitions and the statement of the main results Let A Al . Ak be a partition of n i.e. a non-increasing sequence of positive integers with sum n. We identify the partition A with its Ferrers diagram cf. ECII Section 7.2 . A standard Young tableau is a filling of the shape A with the .