Structures of pairs {a,c} and {b,d} in abelian square-free words ( g 85 ) 2 (a) and ( g 98 ) 2 (a)

© 2005 V. Keränen

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We apply the endomorphism   g 85   (resp. g 98 )  over four letters  { a,b,c,d }  to   g 85 (a)  (resp. g 98 (a)),  where   g 85   and   g 98   are defined (in Mathematica) as follows:

In[1]:=

cyclicPermutation = { a b , b c , c d , d a } ;

g85a = abcacdcbcdcadcdbdabacabadbabcbdbcbacbcdcacbabdabacadcbcdcacdbcbacbcdcacdcbdcdadbdcbca ;

g85b = StringReplace [ g85a , cyclicPermutation ] ;

g85c = StringReplace [ g85b , cyclicPermutation ] ;

g85d = StringReplace [ g85c , cyclicPermutation ] ;

ga = abcacdcbcdcadbdcbdbabcbdcacbabdbabcabdadcdadbdcbdbabdbcbacbcdbabdcdbdcacdbcbacbcdcacdcbdcdadbdcbca ;

gb = StringReplace [ ga , cyclicPermutation ] ;

gc = StringReplace [ gb , cyclicPermutation ] ;

gd = StringReplace [ gc , cyclicPermutation ] ;

{ g98a = ga , g98b = gd , g98c = gc , g98d = gb } ;

Below the (abelian square-free) words   g 85 ( g 85 ( a ) )   and   g 98 ( g 98 ( a ) )   are represented by  85 x 85  and  98 x 98 matrices.  The 85 x 85 matrices are truely symmetric and the 98 x 98 matrices become symmetric, if the letters  b  and  d  are identified.

Generally speaking, our knowledge of the structures of abelian square-free words has not been developing in phase with many other areas of combinatorics on words. We hope that the visualisations below would help in finding efficient new ideas.

( g 85 ) 2 (a)   with all the four  {a,b,c,d} letters:

[Graphics:HTMLFiles/_ac.bd.structures.in.g85.g98_1.gif]

( g 85 ) 2 (a)   with only  {a,c}  letters:

[Graphics:HTMLFiles/_ac.bd.structures.in.g85.g98_2.gif]

( g 85 ) 2 (a)   with the complement  {b,d}  letters:

[Graphics:HTMLFiles/_ac.bd.structures.in.g85.g98_3.gif]

( g 98 ) 2 (a)   with all the four  {a,b,c,d}  letters:

[Graphics:HTMLFiles/_ac.bd.structures.in.g85.g98_4.gif]

( g 98 ) 2 (a)   with only  {a,c}  letters:

[Graphics:HTMLFiles/_ac.bd.structures.in.g85.g98_5.gif]

( g 98 ) 2 (a)   with the complement  {b,d}  letters:

[Graphics:HTMLFiles/_ac.bd.structures.in.g85.g98_6.gif]


Created by Mathematica  (January 17, 2005)