![](/files/happy5.png)
Beneš, Václav E.
Beneš Conjecture (graph-theoretic form) ★★★
Author(s): Beneš
![$ \dag $](/files/tex/2294faf152975267ed7c9a554397e76f5fb394f1.png)
![$ \ell $](/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png)
![$ \diamond $](/files/tex/5649903b740aeedffa3faf46444d03914b80882b.png)
![$ L $](/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png)
![$ 2 $](/files/tex/5271e36bb1c040e0f14061d89cd97d0c86d4e06f.png)
![$ L^m $](/files/tex/3cd6df5ff1e5f14ed88d901c62f9ed57173bf0ed.png)
![$ m\ge1 $](/files/tex/9a0b41ecb89547d72036ec7a219386a491a9153f.png)
![$ L^{2m} $](/files/tex/767ae4a7f4d8c5c33467055f583a03f93843b349.png)
Keywords:
Beneš Conjecture ★★★
Author(s): Beneš
Let be a non-empty finite set. Given a partition
of
, the stabilizer of
, denoted
, is the group formed by all permutations of
preserving each block of
.
![$ \star $](/files/tex/22fb2565a80ccc44dd08308c79b4a6609bfa1d48.png)
![$ {\bf h}_1, \dots, {\bf h}_\ell $](/files/tex/2239f0851ebd51a031f1303512a0df6c50c4e9dc.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ S({\bf h}_1) S({\bf h}_2) \dots S({\bf h}_\ell) $](/files/tex/ab621f0c4f598d3b6709bcd298d88c2106e0c67c.png)
![$ \frak S(E) $](/files/tex/472e24261ccf4c6b4324fae3e02b3e0809bf9992.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ \bf h,\delta(\bf h),\dots,\delta^{\ell-1}(\bf h) $](/files/tex/d3e9972171294f6202bd13a303043160f03f0e58.png)
![$ \bf h $](/files/tex/88933e586874efc507419a53a59c01adad3bba9d.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ \delta $](/files/tex/6468f8f4e0c9f7a9784077872a149582dd4e62fc.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ \bf u $](/files/tex/656698906d536b008577e0fb4874cc00afda605a.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ \varphi $](/files/tex/5211263e826ed3de45d7b004a0e5c468d1a361fd.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ \bf u\wedge\varphi(\bf u)=\bf 0 $](/files/tex/ed184b81383bd05a433a5afc4a7cc4e78211f4bb.png)
![$ \big(\varphi S({\bf u})\big)^{n} $](/files/tex/fe2da412f63dedb4f4e5733f30f6a9f283cb96eb.png)
![$ n\ge2 $](/files/tex/9affc1ab6ec2f6ca13d85ddf4465ad63311fee18.png)
![$$ \frak S(E) = \big(\varphi S({\bf u})\big)^{2n-1}. $$](/files/tex/0d341fdbad6f12fb267f6f04e27ec376ffe3dba6.png)
Keywords:
Shuffle-Exchange Conjecture ★★★
Author(s): Beneš; Folklore; Stone
Given integers , let
be the smallest integer
such that the symmetric group
on the set of all words of length
over a
-letter alphabet can be generated as
(
times), where
is the shuffle permutation defined by
, and
is the exchange group consisting of all permutations in
preserving the first
letters in the words.
![$ d(k,n) $](/files/tex/858410106f855d4731af6a7f19dd3ded70f81b08.png)
![$ d(k,n)=2n-1 $](/files/tex/b83b000fc1d6dd29fb829a03beffef104b4df30e.png)
![$ k,n\ge2 $](/files/tex/1041e1f578aec49a27fbc841578eac3503c9f753.png)
Keywords:
Shuffle-Exchange Conjecture (graph-theoretic form) ★★★
Author(s): Beneš; Folklore; Stone
Given integers , the 2-stage Shuffle-Exchange graph/network, denoted
, is the simple
-regular bipartite graph with the ordered pair
of linearly labeled parts
and
, where
, such that vertices
and
are adjacent if and only if
(see Fig.1).
Given integers , the
-stage Shuffle-Exchange graph/network, denoted
, is the proper (i.e., respecting all the orders) concatenation of
identical copies of
(see Fig.1).
Let be the smallest integer
such that the graph
is rearrangeable.
![$ r(k,n) $](/files/tex/89439aec31f70c82907b7174d89cdee92f6deba0.png)
![$ r(k,n)=2n-1 $](/files/tex/2b890eb205a3c491b93472737bc3c1b09b1fa787.png)
Keywords:
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