On the number of abelian bordered words (with an example of automatic theorem-proving)

In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible symmetric Motzkin paths and paths in $\mathbb{Z}$ not returning to the origin. This study can be extended to abelian unbordered words over an arbitrary alphabet and we derive expressions to compute the number of these words. In particular, over a $3$-letter alphabet, the connection with paths in the triangular lattice is made. Finally, we characterize the lengths of the abelian unbordered factors occurring in the Thue--Morse word using some kind of automatic theorem-proving provided by a logical characterization of the $k$-automatic sequences.