q-Parikh Matrices and q-deformed binomial coefficients of words
<|reference_start|>q-Parikh Matrices and q-deformed binomial coefficients of words: 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 E\u{g}ecio\u{g}lu 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 $u^\omega$, 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 $u^\omega$.   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.<|reference_end|>
@article{renard2024q-parikh,
    title={q-Parikh Matrices and q-deformed binomial coefficients of words},
    author={Antoine Renard, Michel Rigo, Markus A. Whiteland},
    journal={arXiv preprint arXiv:2402.05657},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.05657},
    primaryClass={cs.FL cs.DM math.CO}
}