Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Realizability of Some Combinatorial Sequences

Geng-Rui Zhang

Abstract

A sequence $a=(a_n)_{n=1}^\infty$ of non-negative integers is called realizable if there is a self-map $T:X\to X$ on a set $X$ such that $a_n$ is equal to the number of periodic points of $T$ in $X$ of (not necessarily exact) period $n$, for all $n\geq1$. The sequence $a$ is called almost realizable if there exists a positive integer $m$ such that $(ma_n)_{n=1}^\infty$ is realizable. In this article, we show that certain wide classes of integer sequences are realizable, which contain many famous combinatorial sequences, such as the sequences of Apéry numbers of both kinds, central Delannoy numbers, Franel numbers, Domb numbers, Zagier numbers, and central trinomial coefficients. We also show that the sequences of Catalan numbers, Motzkin numbers, and large and small Schröder numbers are not almost realizable.

Realizability of Some Combinatorial Sequences

Abstract

A sequence of non-negative integers is called realizable if there is a self-map on a set such that is equal to the number of periodic points of in of (not necessarily exact) period , for all . The sequence is called almost realizable if there exists a positive integer such that is realizable. In this article, we show that certain wide classes of integer sequences are realizable, which contain many famous combinatorial sequences, such as the sequences of Apéry numbers of both kinds, central Delannoy numbers, Franel numbers, Domb numbers, Zagier numbers, and central trinomial coefficients. We also show that the sequences of Catalan numbers, Motzkin numbers, and large and small Schröder numbers are not almost realizable.
Paper Structure (4 sections, 9 theorems, 92 equations)

This paper contains 4 sections, 9 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Auxiliary lemmas
  3. Proofs of theorems
  4. Acknowledgments

Key Result

Theorem 13

For every $s\in\mathbb Z_{\geq0}$ and $r\in\mathbb Z^+$, the sequence $(A(n,r,s))_{n=1}^\infty$ is realizable.

Theorems & Definitions (43)

  • Definition 1
  • Remark 2
  • Definition 3
  • Example 4
  • Definition 5
  • Remark 6
  • Definition 7
  • Remark 8
  • Definition 9
  • Remark 10
  • ...and 33 more