The least known length of ordered basis of symmetric group S. A. Kalinchuk, Yu. L. Sagalovich Institute for Information Transmission Problems, Russian.

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Presentation transcript:

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 S n 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 on the set of numbers an ordered system of transpositions of, ordered basis Definition: The system is called ordered basis of symmetric group if any permutation can be represented as 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 of symmetric group is chosen to be equal to ACCT 2008

Main 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 2008

Main 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 2008

Main results Partition Let and Let and be some permutations defined on the set Consider an ordered system of transpositions Example: 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 ACCT 2008 Symmetric group on Partition recurrently the set The system is the order basis of where Let

Ordered basis construction ACCT 2008 Symmetric group on Partition recurrently the set The system is the order basis of where Let

Ordered basis construction ACCT 2008 Symmetric group on Partition recurrently the set The system is the order basis of where Let

Ordered basis construction example ACCT 2008 Since apply

Ordered basis construction example ACCT 2008 Since apply

Ordered basis construction example ACCT 2008

Ordered basis construction example ACCT 2008

Ordered basis construction example ACCT The constructed ordered basis consists of 76 transpositions 120 Total number of all transpositions in S 16 is 120

Ordered basis length ACCT 2008 Differs from lower bound only in factor