Counting with Symmetric Functions
This monograph provides a self-contained introduction to symmetric functions and their use in enumerative combinatorics. It is the first book to explore many of the methods and results that the authors present. Numerous exercises are included throug
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Anthony Mendes Jeffrey Remmel
Counting with Symmetric Functions
Developments in Mathematics VOLUME 43 Series Editors: Krishnaswami Alladi, University of Florida, Gainesville, FL, USA Hershel M. Farkas, Hebrew University of Jerusalem, Jerusalem, Israel
More information about this series at http://www.springer.com/series/5834
Anthony Mendes
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Jeffrey Remmel
Counting with Symmetric Functions
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Anthony Mendes Mathematics Department California Polytechnic State University San Luis Obispo, CA, USA
Jeffrey Remmel Department of Mathematics University of California at San Diego La Jolla, CA, USA
ISSN 1389-2177 ISSN 2197-795X (electronic) Developments in Mathematics ISBN 978-3-319-23617-9 ISBN 978-3-319-23618-6 (eBook) DOI 10.1007/978-3-319-23618-6 Library of Congress Control Number: 2015953218 Mathematics Subject Classification (2010): 05A05, 05E05, 05A15, 05A19, 05E18 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)
Preface
This book is about how symmetric functions can be used in enumeration. The development is entirely self-contained, including an extensive introduction to the ring of symmetric functions. Many of the proofs are combinatorial and involve bijections or sign-reversing involutions. There are numerous exercises with full solutions, many of which highlight interesting mathematical gems. The intended audience is graduate students and researchers in mathematics or related subjects who are interested in counting methods, generating functions, or symmetric functions. The mathematical prerequisites are relatively low; we assume the readers possess a knowledge of elementary combinatorics and linear algebra. We use the basic ideas of group theory and ring theory sparingly in the book, using them mostly in Chapter 6. Chap
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