Properties of Parikh Determinant Under Some Word Operations

The notion of the Parikh matrix was introduced by Mateescu (2001) to study the numerical properties of a word over an alphabet in terms of subwords. The Parikh determinant for a word has recently been introduced to study the analogy of the classical notion of determinant in matrix theory. The Parikh determinant of a word [Formula: see text] over an alphabet [Formula: see text] can be computed by simply finding the number of occurrences of the scattered subword [Formula: see text], i.e., [Formula: see text]. This paper investigates certain Parikh determinant properties in relation to word operations such as partial sum, product, circular variance of words and SShuffle operators etc. A necessary and sufficient condition, for the Parikh determinants of two conjugate ternary words to be equal is also provided.