A note on Pr\"ufer-like coding and counting forests of uniform hypertrees

TL;DR

This paper extends Pr"ufer coding to forests of labeled rooted uniform hypertrees and hypercycles, providing efficient encoding algorithms and counting formulas, including Cayley's and Selivanov's results.

Contribution

It introduces a linear-time encoding and decoding method for hypertree forests, generalizing classical Pr"ufer codes and deriving enumeration formulas.

Findings

Linear-time encoding and decoding algorithms for hypertree forests

Extension of Pr"ufer code to hypergraphs and hypercycles

Derivation of enumeration formulas including Cayley's and Selivanov's

Abstract

This note presents an encoding and a decoding algorithms for a forest of (labelled) rooted uniform hypertrees and hypercycles in linear time, by using as few as $n - 2$ integers in the range $[1,n]$. It is a simple extension of the classical Pr\"{u}fer code for (labelled) rooted trees to an encoding for forests of (labelled) rooted uniform hypertrees and hypercycles, which allows to count them up according to their number of vertices, hyperedges and hypertrees. In passing, we also find Cayley's formula for the number of (labelled) rooted trees as well as its generalisation to the number of hypercycles found by Selivanov in the early 70's.