q-Parikh matrices and q-deformed binomial coefficients of words

Abstract

We have introduced a q-deformation, i.e., a polynomial in q with natural coefficients, of the binomial coefficient of two finite words u and v counting the number of occurrences of v as a subword of u. In this paper, we examine the q-deformation of Parikh matrices as introduced by Eğecioğlu in 2004.

Many classical results concerning Parikh matrices generalize to this new framework: Our first important observation is that the elements of such a matrix are in fact q-deformations of binomial coefficients of words. We also study their inverses and we obtain new identities about q-binomials.

For a finite word z and for the sequence ${\left(p_n\right)}_{n\ge 0}$ of prefixes of an infinite word, we show that the polynomial sequence ${\left(\begin{array}{c}{p}_n\\ z\end{array}\right)}_q$ converges to a formal series. We present links with additive number theory and k-regular sequences. In the case of a periodic word $u^{\omega }$, we generalize a result of Salomaa: the sequence ${\left(\begin{array}{c}{u}^n\\ z\end{array}\right)}_q$ satisfies a linear recurrence relation with polynomial coefficients. Related to the theory of integer partition, we describe the growth and the zero set of the coefficients of the series associated with $u^{\omega }$.

Finally, we show that the minors of a q-Parikh matrix are polynomials with natural coefficients and consider a generalization of Cauchy's inequality. We also compare q-Parikh matrices associated with an arbitrary word with those associated with a canonical word $12\cdots k$ made of pairwise distinct symbols.