Proof of a conjecture on 2-isometric words

Abstract Let F k be the family of the binary words containing exactly k 0s. Ilic, Klavžar and Rho constructed an infinite subfamily of 2-isometric but not 3-isometric words among F 2 . Wei, Yang and Wang further found there are 2-isometric but not 3-isometric words among F k for all k ∈ { 2 , 5 , 6 } and k ≥ 8 , and they conjectured that F 1 , F 3 , F 4 and F 7 are the only families in which there are not 2-isometric but not 3-isometric words. In the present paper, we show that this conjecture is true, and find all the 2-isometric words among F 5 and F 6 .