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\def\volnr{11~(2)}
\def\pubyear{2006}
\def\pubnr{A5}


\Author={Marc A. A. van Leeuwen}
\email={Marc.van-Leeuwen@math.univ-poitiers.fr}
\URL={www-math.univ-poitiers.fr/~maavl/}

\title={Schur functions and alternating sums}

\def\dedication{\medskip\centerline{\boldit
  Dedicated to Richard Stanley on the occasion of his $60\rm^{th}$ birthday}
\medskip}

\MSCcodes={05E05, 05E10}
\keywords={Schur functions, Pieri rule, Murnaghan-Nakayama rule, Jacobi-Trudi
identity, Littlewood-Richardson rule}


\DateSubmitted={Apr 18, 2005}
\DateAccepted={Feb 13, 2006}
\DatePublished={Feb 22, 2006}


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\abstract
 We discuss several well known results about Schur functions that can
 be proved using cancellations in alternating summations; notably we shall
 discuss the Pieri and Murnaghan-Nakayama rules, the Jacobi-Trudi identity and
 its dual (Von N{\"a}gelsbach-Kostka) identity, their proofs using the
 correspondence with lattice paths of Gessel and Viennot, and finally the
 Littlewood-Richardson rule. Our our goal is to show that the mentioned
 statements are closely related, and can be proved using variations of the
 same basic technique. We also want to emphasise the central part that is
 played by matrices over~$\{0,1\}$ and over~$\N$; we show that the
 Littlewood-Richardson rule as generalised by Zelevinsky has elegant
 formulations using either type of matrix, and that in both cases it can be
 obtained by two successive reductions from a large signed enumeration of such
 matrices, where the sign depends only on the row and column sums of the
 matrix.

\maintext


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