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Some counting formulas for $λ$-quiddities over the rings $\mathbb{Z}/2^{m}\mathbb{Z}$

Flavien Mabilat

Abstract

The $λ$-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 their properties are closely linked to the structure and the cardinality of the chosen set. The main objective of this text is to obtain an explicit formula giving the number of $λ$-quiddities of odd size, and a lower and upper bound for the number of $λ$-quiddities of even size, over the rings $\mathbb{Z}/2^{m}\mathbb{Z}$ ($m \geq 2$). We also give explicit formulas concerning the number of $λ$-quiddities of size $n$ over $\mathbb{Z}/8\mathbb{Z}$.

Some counting formulas for $λ$-quiddities over the rings $\mathbb{Z}/2^{m}\mathbb{Z}$

Abstract

The -quiddities of size are -tuples of elements of a fixed set, solutions of a matrix equation appearing in the study of Coxeter's friezes. Their number and their properties are closely linked to the structure and the cardinality of the chosen set. The main objective of this text is to obtain an explicit formula giving the number of -quiddities of odd size, and a lower and upper bound for the number of -quiddities of even size, over the rings (). We also give explicit formulas concerning the number of -quiddities of size over .
Paper Structure (6 sections, 14 theorems, 42 equations)

This paper contains 6 sections, 14 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Proofs of the counting formulas
  3. Preliminary results
  4. Proof of theorem \ref{['prin1']}
  5. The case of $\mathbb{Z}/8\mathbb{Z}$
  6. Numerical applications

Key Result

Theorem 1.1

Let $q$ be the power of a prime number $p$ and $n>4$. i) If $n$ is odd then $u_{n,q}^{-}=\left[\frac{n-1}{2}\right]_{q^{2}}$. ii) If $n$ is even then there exists $m \in \mathbb{N^{*}}$ such that $n=2m$.

Theorems & Definitions (22)

  • Theorem 1.1: Morier-Genoud, Mo2 Theorem 1
  • Theorem 1.2: CM Theorem 1.1
  • Theorem 1.3: CM Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1: CM lemma 2.16 and proposition 2.18
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 12 more