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{\bf Catalin Zara }
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{\bf Chains, Subwords, and Fillings: Strong Equivalence of Three Definitions of the Bruhat Order}
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Let $S_n$ be the group of permutations of $[n]=\{1,\ldots,n\}$. The
Bruhat order on $S_n$ is a partial order relation, for which there
are several equivalent definitions. Three well-known conditions are
based on ascending chains, subwords, and comparison of matrices,
respectively. We express the last using fillings of tableaux, and
prove that the three equivalent conditions are satisfied {\em in the
same number of ways}.
\bye