\magnification=1440
\font\bigtenrm=cmr10 scaled\magstep4
\def\binom#1#2{{#1{#2\atopwithdelims()}}}
\def\dbinom{\protect\frbinom\displaystyle}
\def\tbinom{\protect\frbinom\textstyle}
Abstract for 
Anders Bj\"orner and Francesco Brenti,
An improved \break tableau criterion for Bruhat order

To decide whether two permutations are comparable in Bruhat order of
$S_n$ with the well-known tableau criterion requires $\binom{n}{2}$
comparisons of entries in certain sorted arrays. We show that to
decide whether $x\le y$ only $d_1+d_2+...+d_k$ of these comparisons
are needed, where $\{d_1,d_2,...,d_k\} = \{i|x(i)>x(i+1)\}$. This is
obtained as a consequence of a sharper version of Deodhar's criterion, 
which is valid for all Coxeter groups. 


\bye