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Consider the following list of

alphabetic sequences:


ABCDEFG
ABCEGFD
ACGFEBD
ADEBFGC
AEGFBCD
AGBCEDF
AGFCEBD
BAFECDG
BCFEDAG
BDACFEG
BDAGCEF
BFDACGE
BFEACDG
BGCDEFA
CAEBFGD
CAEFDGB
CBFAGDE
CDAEFBG
CDGEFAB
CEDGFAB
CFDGAEB
DAECGBF
DBECAGF
DECBAGF
DEGBAFC
DFAGCEB
DGBAECF
DGBFACE
EAGCDBF
EBCFGAD
EBDFGCA
ECFAGDB
ECFGBDA
EFBDGCA
EGBDCFA
FABGDEC
FAEGDCB
FCDEBAG
FDABEGC
FEGCDBA
FEGDABC
FGABDEC
GBCAFDE
GCADBFE
GCFDBEA
GDCABFE
GEBFACD
GFDBCAE
GFDEBAC
[/code]

They are listed in alphabetic order

-- [b]NOT[/b] in the order in which they

were generated. These were generated

using a pair of 7-cycle permutations,

P and Q, on 7 objects. Each of these

49 sequences was produces by applying

P[sup]i[/sup]Q[sup]j[/sup] to ABCDEFG for all pairs i and j

such that 0 < i,j < 8. Your task is

to:

1. find such a P and Q which will

produce all the sequences.

2. determine how many equivalent

pairs of such permutations

there are.

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Consider the following list of

alphabetic sequences:


ABCDEFG

ABCEGFD

ACGFEBD

ADEBFGC

AEGFBCD

AGBCEDF

AGFCEBD

BAFECDG

BCFEDAG

BDACFEG

BDAGCEF

BFDACGE

BFEACDG

BGCDEFA

CAEBFGD

CAEFDGB

CBFAGDE

CDAEFBG

CDGEFAB

CEDGFAB

CFDGAEB

DAECGBF

DBECAGF

DECBAGF

DEGBAFC

DFAGCEB

DGBAECF

DGBFACE

EAGCDBF

EBCFGAD

EBDFGCA

ECFAGDB

ECFGBDA

EFBDGCA

EGBDCFA

FABGDEC

FAEGDCB

FCDEBAG

FDABEGC

FEGCDBA

FEGDABC

FGABDEC

GBCAFDE

GCADBFE

GCFDBEA

GDCABFE

GEBFACD

GFDBCAE

GFDEBAC

They are listed in alphabetic order

-- NOT in the order in which they

were generated. These were generated

using a pair of 7-cycle permutations,

P and Q, on 7 objects. Each of these

49 sequences was produces by applying

PiQj to ABCDEFG for all pairs i and j

such that 0 < i,j < 8. Your task is

to:

1. find such a P and Q which will

produce all the sequences.

2. determine how many equivalent

pairs of such permutations

there are.

Interesting problem

Let p be the permutation (7,5,2,6,1,3,4) and q be the permutation (7,6,4,5,2,1,3). The pair P and Q that satisfy the OP can be generated by

P = pi

Q = qj; i, j = 1,2,..,6

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