# 2026
PDF (2.1MB)
(A144959) 2026 hyperforests spanning 10 unlabeled nodes without isolated vertices.
# 1 Explanation
A graph is collection of nodes connected with edges.
Nodes can be labeled (in which case you can tell them apart) or unlabeled (in which case all nodes are equivalent).
In a normal graph, edges connect 2 nodes.
A tree is a connected acyclic graph.
Connected means that every node is reachable from every other node by a path.
A path is a sequence of edges, with each next edge starting from the end of the previous edge.
Acyclic means there are no cycles.
A cycle is path that doesn’t repeat any edge, that ends up where it starts.
A forest is a collection of trees.
An isolated node is not connected by any edge.
In a hypergraph, hyperedges can connect any number of nodes.
A hypergraph can be converted to its incidence graph: turn each hyperedge into another node, and make the graph have edges between the original nodes and the new hyperedge node whenever the original node is in that hyperedge.
This new graph is bipartite: there are two parts, the original nodes, and the hyperedge nodes, and there are only edges between the parts and not within the parts.
A hyperforest is a hypergraph whose incidence graph is a forest.
Two trees are isomorphic if they have the same shape. See tree isomorphism.
Two forests are isomorphic if they have the same number of trees and those trees can be paired up isomorphically.
Two hyperforests are isomorphic if their incidence graphs are isomorphic forests.
A fun problem is counting isomorphism classes.
An isomorphism class is all the items that are mutually isomorphic.
In the Online Encyclopedia of Integer Sequences, “the number of X” usually means “the number of isomorphism classes of X”.
The number of hyperforests spanning n unlabeled nodes without isolated vertices is counted in oeis.org/A144959.
When n = 10, the number is 2026.
# 2 Source Code
git clone https://code.mathr.co.uk/oeis-diagrams.git
cd oeis-diagrams/2026
A144959.cc is the main source code in C++23.
A144959.sh compiles and runs, generating over 4000 files.
Final output is in 10.svg, which I printed to PDF using Firefox.