A Matrix Q-Analogue of the Parikh Map
Ömer Eğeci̇oğlu, Óscar H. Ibarra · Kluwer Academic Publishers eBooks · 2006
We introduce an extension of the Parikh mapping called the Parikh q-matrix mapping, which takes its values in matrices with polynomial entries. The morphism constructed represents a word ω over a κ-letter alphabet as a κ-dimensional upper-triangular matrix with entries that are nonnegative integral polynomials in variable q. We show that by appropriately embedding the κ-letter alphabet into the κ+1)-letter alphabet and putting q=1, we obtain the extension of the Parikh mapping to (κ+1)-dimensional (numerical)matrices introduced by Mateescu, Salomaa, Salomaa, and Yu. The Parikh q-matrix mapping however, produces matrices that carry more information about ω than the numerical Parikh matrix. The entries of the q-matrix image of ω under this morphism is constructed by the q-counting number of occurrences of certain words as scattered subwords of ω