Y-frieze patterns

Motivated by cluster ensembles, we introduce a new variant of frieze patterns associated to acyclic cluster algebras, which we call $\mathbf{Y}\mathit{\text{-frieze}}\mathit{\text{patterns}}$. Using the mutation rules for $\mathbf{Y}$-variables, we define a large class of $\mathbf{Y}$-frieze patterns called $\mathit{\text{unitary}}\mathbf{Y}\mathit{\text{-frieze}}\mathit{\text{patterns}}$, and show that the ensemble map induces a map from (unitary) frieze patterns to (unitary) $\mathbf{Y}$-frieze patterns. In rank 2, we show that $\mathbf{Y}$-frieze patterns are (associated to) friezes of generalised cluster algebras. In finite type (not necessarily rank 2), we show that $\mathbf{Y}$-frieze patterns share the same symmetries as frieze patterns, and prove that their number is finite.