A Matrix q-Analogue of the Parikh Map

Abstract 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 £