Realizability of Some Combinatorial Sequences

TL;DR

This paper investigates which combinatorial sequences can be represented as counts of periodic points of a self-map, establishing realizability for many famous sequences and non-realizability for others.

Contribution

It proves that many well-known combinatorial sequences are realizable as periodic point counts, while others are not almost realizable, expanding understanding of sequence realizability.

Findings

Many classical combinatorial sequences are realizable.

Sequences like Catalan and Motzkin are not almost realizable.

The paper characterizes classes of sequences based on realizability.

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 Ap\'ery 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 Schr\"oder numbers are not almost realizable.