A MATRIX Q-ANALOGUE OF THE PARIKH MAP

摘要

We introduce an extension of the Parikh mapping called the Parikh q-matrix mapping, which takes its values in matrices with polynomial entries. The mor-phism constructed represents a word w over a k-letter alphabet as a k-dimensional upper-triangular matrix with entries that are nonnegative integral polynomials in variable q. We show that by appropriately embedding the k-letter alphabet into the (k + 1)-letter alphabet and putting q = 1, we obtain the extension of the Parikh mapping to (k +1)-dimensional (numerical) matrices introduced by Ma-teescu, Salomaa, Salomaa, and Yu. The Parikh q-matrix mapping however, produces matrices that carry more information about w than the numerical Parikh matrix. The entries of the q-matrix image of w under this morphism is constructed by q-counting the number of occurrences of certain words as scattered subwords of w.