the least known length of ordered basis of symmetric group s. a. kalinchuk, yu. l. sagalovich...
TRANSCRIPT
The least known length The least known length of ordered basis of of ordered basis of symmetric groupsymmetric group
S. A. Kalinchuk, S. A. Kalinchuk, Yu. L. SagalovichYu. L. Sagalovich
Institute for Information Transmission Institute for Information Transmission Problems, Russian Academy of SciencesProblems, Russian Academy of Sciences
ACCT 2008ACCT 2008
IntroductionIntroduction
ACCT 2006 paper “The problem of minimal ordered basis of symmetric group”
Set of all transpositions as a basis of symmetric group Sn
Questions Is it possible to use less number of transpositions for
obtaining all n! permutations? Is it possible to fix the sequence of transpositions by the
only way for all products? (2,4) (2,3) (1,4) (1,2) (1,3)
ACCT 2008ACCT 2008
Ordered basis definitionOrdered basis definition
symmetric group with degree on the set of numbers
an ordered system of transpositions of ,
Definition:The system is called ordered basisordered basis of symmetric group ifany permutation can be represented as
where
ACCT 2008ACCT 2008
PreliminariesPreliminaries
There exist the ordered bases with the transpositions’ number of order . For example,
The obtained result is based on that the degree of symmetric group is chosen to be equal to
ACCT 2008ACCT 2008
Main resultsMain results Let , Partition Proposition 1:
Any permutation of group can be factored as
where and are some permutations belonging to symmetric groups and correspondingly, and a permutation of group has the form as
Example:
ACCT 2008ACCT 2008
Main resultsMain results
Proposition 2:
Let and be ordered bases of groups and correspondingly.
Let be an ordered system of transpositions of group , and let this system generate permutations of the form .
Then the system
is the ordered basis of group
ACCT 2008ACCT 2008
Main resultsMain results
Partition
Let and
Let and be some permutations defined on the set Consider an ordered system of transpositions
Example:
ACCT 2008ACCT 2008
Main resultsMain results
Proposition 3:
Let and be some ordered systems of transpositions generating permutations of the forms and correspondingly.
Then the system
generates permutations of the form at any and .
ACCT 2008ACCT 2008
Ordered basis Ordered basis constructionconstruction
ACCT 2008ACCT 2008
Symmetric group on Partition recurrently the set The system
is the order basis of where Let
Ordered basis Ordered basis constructionconstruction
ACCT 2008ACCT 2008
Symmetric group on Partition recurrently the set The system
is the order basis of where Let
Ordered basis Ordered basis constructionconstruction
ACCT 2008ACCT 2008
Symmetric group on Partition recurrently the set The system
is the order basis of where Let
Ordered basis Ordered basis construction exampleconstruction example
ACCT 2008ACCT 2008
Since
apply
Ordered basis Ordered basis construction exampleconstruction example
ACCT 2008ACCT 2008
Since
apply
Ordered basis Ordered basis construction exampleconstruction example
ACCT 2008ACCT 2008
Ordered basis Ordered basis construction exampleconstruction example
ACCT 2008ACCT 2008
Ordered basis Ordered basis construction exampleconstruction example
ACCT 2008ACCT 2008
The constructed ordered basis consists of7676 transpositions
Total number of all transpositions in S16 is 120120
Ordered basis Ordered basis lengthlength
ACCT 2008ACCT 2008
Differs from lower bound
only in factor