Parikh matrices and amiable words

Using the fact that the Parikh matrix mapping is not an injective mapping, the paper investigates some properties of the set of words with the same Parikh matrix; these words are called “amiable”. The presented results extend the results obtained in [A. Atanasiu, Binary amiable words, Int. J. Found. Comput. Sci. 18 (2) (2007) 387–400] for the binary case. In particular it is shown that all the words having the same Parikh matrix can be obtained one from another by applying only two types of transformations. Moreover, the mirrors of two amiable words are also amiable (thus forming a symmetrical class of words).