Geodesic
An improved tableau criterion for Bruhat order
The electronic journal of combinatorics, Tome 3 (1996) no. 1
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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.
DOI : 10.37236/1246
Classification : 20F55, 14M15
Mots-clés : permutations, Bruhat order, tableau criterion, Deodhar's criterion, Coxeter groups 
@article{10_37236_1246,
     author = {Anders Bj\"orner and Francesco Brenti},
     title = {An improved tableau criterion for {Bruhat} order},
     journal = {The electronic journal of combinatorics},
     year = {1996},
     volume = {3},
     number = {1},
     doi = {10.37236/1246},
     zbl = {0884.05096},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1246/}
}
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Anders Björner; Francesco Brenti. An improved tableau criterion for Bruhat order. The electronic journal of combinatorics, Tome 3 (1996) no. 1. doi: 10.37236/1246

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