Generalized Parikh Matrices For Tracking Subsequence Occurrences

TL;DR

This paper introduces a generalized Parikh matrix framework that captures more detailed subsequence occurrence information while maintaining key algebraic properties, extending the original concept with new matrix mappings and theoretical insights.

Contribution

It proposes a new generalized Parikh matrix based on subsequence tracking, connecting it to existing matrices and analyzing its algebraic properties.

Findings

Certain minors of the generalized matrices have nonnegative determinants

Generalized subword histories are equivalent to linear ones

The new framework retains homomorphic properties of original Parikh matrices

Abstract

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.