A Matrix q-Analogue of the Parikh Map

We introduce an extension of the Parikh mapping called the Parikh -matrix mapping, which takes its values in matrices with polynomial entries. The morphism constructed represents a word over a -letter alphabet as a -dimensional upper-triangular matrix with entries that are nonnegative integral polynomials in variable . We show that by appropriately embedding the -letter alphabet into the -letter alphabet and putting , we obtain the extension of the Parikh mapping to -dimensional (numerical) matrices introduced by Mateescu, Salomaa, Salomaa, and Yu. The Parikh -matrix mapping however, produces matrices that carry more information about than the numerical Parikh matrix. The entries of the -matrix image of under this morphism is constructed by -counting the number of occurrences of certain words as scattered subwords of .