On the Number of Fixed-Length Semiorders

TL;DR

This paper establishes a bijection between semiorders of fixed length and ordered trees, enabling enumeration through generating functions and explicit formulas, advancing combinatorial understanding of semiorders.

Contribution

It introduces a bijection linking semiorders to ordered trees, facilitating enumeration and structural analysis of semiorders of fixed length.

Findings

Derived generating functions for semiorders

Explicit formulas for labeled and unlabeled semiorders

Recurrence relations and combinatorial proofs

Abstract

A semiorder is a partially ordered set $P$ with two certain forbidden induced subposets. This paper establishes a bijection between $n$-element semiorders of length $H$ and $(n+1)$-node ordered trees of height $H+1$. This bijection preserves not only the number of elements, but also much additional structure. Based on this correspondence, we calculate the generating functions and explicit formulas for the numbers of labeled and unlabeled $n$-element semiorders of length $H$. We also prove several concise recurrence relations and provide combinatorial proofs for special cases of the explicit formulas.