A note on Prüfer-like coding and counting forests of uniform hypertrees

The present note is designing encoding and decoding algorithms for a forest of rooted uniform hypertrees and hypercycles in linear time, by using as little as n− 2 integers in the range [1, n]. This simple extension of the classical Prüfer code for rooted trees to an encoding for forests of rooted uniform hypertrees and hypercycles makes it possible to count them up, according to their number of vertices, hyperedges and hypertrees. In passing, we find Cayley’s formula for the number of rooted trees as well as its generalisation to the number of hypercycles found by Selivanov in the early 70’s