Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
The least known length of ordered basis of symmetric group S. A. Kalinchuk, Yu. L. Sagalovich Institute for Information Transmission Problems, Russian Academy of Sciences ACCT 2008 Introduction 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 2008 Ordered basis definition symmetric group with degree an ordered system of transpositions of on the set of numbers Definition: The system is called ordered basis of symmetric group any permutation can be represented as , if where ACCT 2008 Preliminaries There exist the ordered bases with the transpositions’ number of order . For example, The obtained result is based on that the degree symmetric group is chosen to be equal to of ACCT 2008 Main results Let , Partition Proposition 1: Any permutation where groups of group can be factored as and are some permutations belonging to symmetric and correspondingly, and a permutation of group has the form as Example: ACCT 2008 Main results Proposition 2: Let and be ordered bases of groups correspondingly. and 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 2008 Main results Partition Let Let and be some permutations defined on the set Consider an ordered system of transpositions Example: and ACCT 2008 Main 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 2008 Ordered basis construction Symmetric group on Partition recurrently the set The system is the order basis of Let where ACCT 2008 Ordered basis construction Symmetric group on Partition recurrently the set The system is the order basis of Let where ACCT 2008 Ordered basis construction Symmetric group on Partition recurrently the set The system is the order basis of Let where ACCT 2008 Ordered basis construction example Since apply ACCT 2008 Ordered basis construction example Since apply ACCT 2008 Ordered basis construction example ACCT 2008 Ordered basis construction example ACCT 2008 Ordered basis construction example The constructed ordered basis consists of 76 transpositions Total number of all transpositions in S16 is 120 ACCT 2008 Ordered basis length Differs from lower bound only in factor ACCT 2008