Introducing q-deformed binomial coefficients of words

Gaussian binomial coefficients are q-analogues of the binomial coefficients of integers. On the other hand, binomial coefficients have been extended to finite words, i.e., elements of a finitely generated free monoid. In this paper, we bring these two notions together by introducing q-analogues of binomial coefficients of words. We study their basic properties, e.g., by extending classical formulas such as the q-Vandermonde and Manvel–Meyerowitz–Schwenk–Smith–Stockmeyer identities to our setting. These q-deformations contain much richer information than the original coefficients. From an algebraic perspective, we introduce a q-shuffle and a family of q-infiltration products for non-commutative formal power series. Finally, we apply our results to generalize a theorem of Eilenberg characterizing so-called p-group languages. We show that a language is of this type if and only if it is a Boolean combination of specific languages defined through q-binomial coefficients seen as polynomials over Fp.