Geodesic
Parametrization of the Solutions of the Equation $x_1x_2\dots x_{n-1}x_n=x_nx_{n-1}\dots x_2x_1$ in a Free Monoid
Matematičeskie zametki, Tome 89 (2011) no. 6, pp. 879-884

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A parametrizing function $\mathrm{Sm}$ is introduced. The parametrizing function is a recursive function depending on lexicographic variables, natural variables, and variables whose values are finite sequences of natural variables. Using the function $\mathrm{Sm}$, we construct formulas that provide all the solutions of the equation $$ x_1x_2\dots x_{n-1}x_n=x_nx_{n-1}\dots x_2x_1 $$ in a free monoid $\langle a_1,a_2,\dots,a_\omega\rangle$ and only them.
Keywords: free monoid, parametrizing function, recursive function, lexicographic variable, list of words. 
@article{MZM_2011_89_6_a7,
     author = {G. S. Makanin},
     title = {Parametrization of the {Solutions} of the {Equation} $x_1x_2\dots x_{n-1}x_n=x_nx_{n-1}\dots x_2x_1$ in a {Free} {Monoid}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {879--884},
     publisher = {mathdoc},
     volume = {89},
     number = {6},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_89_6_a7/}
}
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G. S. Makanin. Parametrization of the Solutions of the Equation $x_1x_2\dots x_{n-1}x_n=x_nx_{n-1}\dots x_2x_1$ in a Free Monoid. Matematičeskie zametki, Tome 89 (2011) no. 6, pp. 879-884. http://geodesic.mathdoc.fr/item/MZM_2011_89_6_a7/