SOME COUNTING FORMULAE FOR -QUIDDITIES OVER THE RINGS

Abstract

Abstract The $\lambda $ -quiddities of size n are n-tuples of elements of a fixed set, solutions of a matrix equation appearing in the study of Coxeter’s friezes. Their number and properties are closely linked to the structure and the cardinality of the chosen set. Our main objective is an explicit formula giving the number of $\lambda $ -quiddities of odd size, and a lower and upper bound for the number of $\lambda $ -quiddities of even size, over the rings ${\mathbb {Z}}/2^{m}{\mathbb {Z}}$ ( $m \geq 2$ ). We also give explicit formulae for the number of $\lambda $ -quiddities of size n over ${\mathbb {Z}}/8{\mathbb {Z}}$ .