Algebraic Generating Functions for Languages Avoiding Riordan Patterns

We study the languages $\mathfrak{L}^{[\mathfrak{p}]}\subset \{0,1\}^*$ of binary words $w$ avoiding a given pattern $\mathfrak{p}$ such that $|w|_0\ leq |w|_1$ for every $w\in \mathfrak{L}^{[\mathfrak{p}]},$ where $|w|_0$ and $|w|_1$ correspond to the number of $0$-bits and $1$-bits in the word $w$, respectively. In particular, we concentrate on patterns $\mathfrak{p}$ related to the concept of Riordan arrays. These languages are not regular and can be enumerated by algebraic generating functions corresponding to many integer sequences which are unknown in the On-Line Encyclopedia of Integer Sequences. We give explicit formulas for these generating functions expressed in terms of the autocorrelation polynomial of $\mathfrak{p}$ and also give explicit formulas for the coefficients of some particular patterns, algebraically and combinatoriall