  We introduce and study a generalized Parikh matrix mapping based on tracking
the occurrence counts of special types of subsequences. These matrices retain
more information about a word than the original Parikh matrix mapping while
preserving the homomorphic property. We build the generalization by first
introducing the Parikh factor matrix mapping and extend it to the Parikh
sequence matrix mapping. We establish an interesting connection between the
generalized Parikh matrices and the original ones and use it to prove that
certain important minors of a Parikh sequence matrix have nonnegative
determinant. Finally, we generalize the concept of subword histories and show
that each generalized subword history is equivalent to a linear one.
