Structures of pairs {a,c} and {b,d} in abelian square-free words ${\left({g}_{85}\right)}^{2}$(a) and ${\left({g}_{98}\right)}^{2}$(a)

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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:=

$\mathrm{cyclicPermutation}=\left\{"a"\to "b","b"\to "c","c"\to "d","d"\to "a"\right\};$

$\mathrm{g85a}="abcacdcbcdcadcdbdabacabadbabcbdbcbacbcdcacbabdabacadcbcdcacdbcbacbcdcacdcbdcdadbdcbca";$

$\mathrm{g85b}=\mathrm{StringReplace}\left[\mathrm{g85a},\mathrm{cyclicPermutation}\right];$

$\mathrm{g85c}=\mathrm{StringReplace}\left[\mathrm{g85b},\mathrm{cyclicPermutation}\right];$

$\mathrm{g85d}=\mathrm{StringReplace}\left[\mathrm{g85c},\mathrm{cyclicPermutation}\right];$

$\mathrm{ga}="abcacdcbcdcadbdcbdbabcbdcacbabdbabcabdadcdadbdcbdbabdbcbacbcdbabdcdbdcacdbcbacbcdcacdcbdcdadbdcbca";$

$\mathrm{gb}=\mathrm{StringReplace}\left[\mathrm{ga},\mathrm{cyclicPermutation}\right];$

$\mathrm{gc}=\mathrm{StringReplace}\left[\mathrm{gb},\mathrm{cyclicPermutation}\right];$

$\mathrm{gd}=\mathrm{StringReplace}\left[\mathrm{gc},\mathrm{cyclicPermutation}\right];$

$\left\{\mathrm{g98a}=\mathrm{ga},\mathrm{g98b}=\mathrm{gd},\mathrm{g98c}=\mathrm{gc},\mathrm{g98d}=\mathrm{gb}\right\};$

Below the (abelian square-free) words  ${g}_{85}\left({g}_{85}\left(a\right)\right)$  and  ${g}_{98}\left({g}_{98}\left(a\right)\right)$  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.

${\left({g}_{85}\right)}^{2}$(a)   with all the four  {a,b,c,d} letters: ${\left({g}_{85}\right)}^{2}$(a)   with only  {a,c}  letters: ${\left({g}_{85}\right)}^{2}$(a)   with the complement  {b,d}  letters: ${\left({g}_{98}\right)}^{2}$(a)   with all the four  {a,b,c,d}  letters: ${\left({g}_{98}\right)}^{2}$(a)   with only  {a,c}  letters: ${\left({g}_{98}\right)}^{2}$(a)   with the complement  {b,d}  letters: Created by Mathematica  (January 17, 2005)