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Self-descriptive Sequences directed by two Periodic Sequences

Shigeki Akiyama, Damien Jamet, Irène Marcovici, Mai-Linh Trân-Công

TL;DR

A class of self-descriptive sequences that can be explicitly computed and whose frequencies are known are exhibited and it is proved that the sequence introduced in \citeBJM23 has the expected frequencies of occurrences.

Abstract

In the present work, we exhibit a class of self-descriptive sequences that can be explicitly computed and whose frequencies are known. In particular, as a corollary of our main result, we prove that the sequence introduced in \citeBJM23 has the expected frequencies of occurrences.

Self-descriptive Sequences directed by two Periodic Sequences

TL;DR

A class of self-descriptive sequences that can be explicitly computed and whose frequencies are known are exhibited and it is proved that the sequence introduced in \citeBJM23 has the expected frequencies of occurrences.

Abstract

In the present work, we exhibit a class of self-descriptive sequences that can be explicitly computed and whose frequencies are known. In particular, as a corollary of our main result, we prove that the sequence introduced in \citeBJM23 has the expected frequencies of occurrences.

Paper Structure

This paper contains 4 sections, 2 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Definitions and basic notions
  3. Main result
  4. Conclusion and perspectives

Key Result

Theorem 1

Let $u \in \{1,2\}^\mathbb{N}$ be a sequence over $\{1,2\}$ directed by two periodic sequences $T_1 = (x_1)^\omega \in \{1,2\}^\mathbb{N}$ and $T_2 = (x_2)^\omega \in \{1,2\}^\mathbb{N}$, with $x_1,x_2 \in \{1,2\}^*$. Let $p_1 = \dfrac{|x_1|_1}{|x_1|}$ and $q_2 = \dfrac{|x_2|_2}{|x_2|}$. One has with $\Delta = (p_1 + 2q_2)^2 - 8(p_1 + q_2 - 1)$. If $\delta = (\delta_n)_{n \in \mathbb{N}} \in \{1,

Theorems & Definitions (6)

  • Definition 1: Self-descriptive sequence
  • Theorem 1: Main result
  • proof : Sketch of proof
  • Definition 2: BJM sequence BJM23
  • Corollary 1
  • proof