[en] 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.
Disciplines :
Mathematics Computer science
Author, co-author :
Goc, Daniel
Rampersad, Narad
Rigo, Michel ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Salimov, Pavel
Language :
English
Title :
On the number of abelian bordered words (with an example of automatic theorem-proving)
Publication date :
2014
Journal title :
International Journal of Foundations of Computer Science
ISSN :
0129-0541
Publisher :
World Scientific Publishing Company
Volume :
8
Pages :
1097-1110
Peer reviewed :
Peer Reviewed verified by ORBi
Commentary :
This is an extended version of the conference paper.
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