SOME COUNTING FORMULAE FOR -QUIDDITIES OVER THE RINGS

Pub Date : 2024-05-16 DOI:10.1017/s0004972724000340

FLAVIEN MABILAT

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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

${\mathbb {Z}}/8{\mathbb {Z}}$

.

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环上的-奇数的一些计算公式

大小为 n 的 $\lambda $ -quiddities 是一个固定集合的 n 个元素元组，是考克赛特门楣研究中出现的矩阵方程的解。它们的数量和性质与所选集合的结构和万有引力密切相关。我们的主要目标是给出奇数大小的$\lambda $ -quiddities的明确公式，以及偶数大小的$\lambda $ -quiddities的下限和上限，它们都在${mathbb {Z}}/2^{m}{mathbb {Z}}$ （$m \geq 2$）环上。我们还给出了在 ${mathbb {Z}}/8{mathbb {Z}}$ 上大小为 n 的 $\lambda $ -quiddities 的明确公式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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