q-Parikh Matrices and q-deformed binomial coefficients of words
Antoine Renard, Michel Rigo, Markus A. Whiteland
TL;DR
This work introduces a robust $q$-deformation framework for Parikh matrices, defining $q$-Parikh matrices $\mathcal{P}_z(w)$ that encode subword information through $q$-binomial coefficients and extend Eğecioğlu's approach via Şerbănuţă's construction. It develops inverse formulas (under a no-$aa$ condition) linked to reversal, derives new $q$-binomial identities, and demonstrates convergence of $\binom{p_n}{z}_{q}$ to formal power series with automatic/regular structure in the periodic and automatic cases. The paper further proves that minors of $\mathcal{P}_z(u)$ have nonnegative coefficients, establishes $q$-analogues of Cauchy-type inequalities, and shows how generalized $q$-Parikh matrices can be expressed in terms of Eğecioğlu's canonical words, unifying several strands of combinatorics, number theory, and formal power series. These results provide new algebraic and combinatorial tools for studying word substructures, subword statistics, and their $q$-deformations, with connections to partition theory and automatic sequences.
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 as an application, we obtain new identities about q-binomials. For a finite word z and for the sequence $(p_n)_{n\ge 0}$ of prefixes of an infinite word, we show that the polynomial sequence $\binom{p_n}{z}_q$ converges to a formal series. We present links with additive number theory and k-regular sequences. In the case of a periodic word $u^ω$, we generalize a result of Salomaa: the sequence $\binom{u^n}{z}_q$ 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 $u^ω$. 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.