tailieunhanh - Báo cáo toán học: "Lyndon words and transition matrices between elementary, homogeneous and monomial symmetric functions"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Lyndon words and transition matrices between elementary, homogeneous and monomial symmetric functions. | Lyndon words and transition matrices between elementary homogeneous and monomial symmetric functions Andrius Kulikauskas Jeffrey Remmel Minciu Sodas Laboratory Department of Mathematics Vilnius Lithuania University of California San Diego La Jolla CA 92093-0112. USA La Jolla CA 92093-0112. USA ms@ jremmel@ Submitted Jun 23 2004 Accepted Feb 22 2006 Published Feb 28 2006 MR Subject Classification 05E05 05A99 Abstract Let hA eA and mA denote the homogeneous symmetric function the elementary symmetric function and the monomial symmetric function associated with the partition A respectively. We give combinatorial interpretations for the coefficients that arise in expanding mA in terms of homogeneous symmetric functions and the elementary symmetric functions. Such coefficients are interpreted in terms of certain classes of bi-brick permutations. The theory of Lyndon words is shown to play an important role in our interpretations. 1 Introduction Let An denote the space of homogeneous symmetric functions of degree n in infinitely many variables X1 x2 . There are six standard bases of An mA A-n the monomial symmetric functions hA A-n the complete homogeneous symmetric functions eA A-n the elementary symmetric functions pA A-n the power symmetric functions sA A-n the Schur functions and fA A-n the forgotten symmetric functions where A F n denotes that A is a partition of n. We let Ế A denote the length of A . A equals the number of parts of A. The entries of the transition matrices between these bases of symmetric functions all have combinatorial significance. For example Doubilet 2 showed that all such entries could be interpreted via the lattice of set partitions X and The authors would like to thank the anonymous referee who suggested numerous improvements for the presentation of this paper. Supported in part by NSF grant DMS 0400507 THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R18 1 its Mobius function. More recently Beck Remmel and Whitehead 1 gave a