Profile and hereditary classes of ordered relational structures

TL;DR

This paper explores the profile and generating functions of classes of finite ordered relational structures, extending known results from permutation classes and examining hereditary well quasi order properties.

Contribution

It extends results on algebraic generating functions from permutation classes to ordered binary relational structures, focusing on hereditary well quasi orders.

Findings

Generated algebraic series for classes of ordered structures

Connected hereditary well quasi order properties to generating functions

Addressed open questions on hereditary classes

Abstract

Let $\mathfrak{C}$ be a class of finite combinatorial structures. The \textit{profile} of $\mathfrak{C}$ is the function $\varphi_{\mathfrak{C}}$ which counts, for every integer $n$, the number $\varphi_{\mathfrak{C}}(n)$ of members of $\mathfrak{C}$ defined on $n$ elements, isomorphic structures been identified. The \textit{generating function of} $\mathfrak{C}$ is $\mathcal {H}_{\mathfrak{C}}(x):=\sum_{n\geqq 0}\varphi_{\mathfrak{C}}(n)x^{n}$. Many results about the behavior of the function $\varphi_{\mathfrak{C}}$ have been obtained. Albert and Atkinson have shown that the generating series of several classes of permutations are algebraic. In this paper, we show how their results extend to classes of ordered binary relational structures; putting emphasis on the notion of hereditary well quasi order, we discuss some of their questions and answer one.