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What positive integer can be expressed any of these ways?
\(2 \times 2013\)\(45^2+1\)\(3 \times 26^2 - 2\)\(2^{11} - 2 \times 11\)\(505+506+507+508\)\(\displaystyle\left(45+\cfrac{1}{90 + \cfrac{1}{90+ \cfrac{1}{90 + \dotsb}}}\right)^2\) - Zeckendorf Representation What positive integer can be uniquely expressed as a sum of Fibonacci numbers as \(1597 + 377 + 34 + 13 + 5?\)
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Hyperforests A graph is a set of nodes equipped with a set of edges that each connect a pair of distinct nodes. A tree is a graph that has no cycles, no subset of edges that forms a loop. Graphs don’t have to be entirely connected; a forest is a graph comprised of one or more disjoint trees.
A hypergraph is a generalization of a graph where “edges” may now go between more than two vertices; these generalized edges are called hyperedges. Essentially a hypergraph on \(n\) nodes is any subset of the powerset of \(\{1,\dotsc, n\}\) without singleton elements. A hypergraph is a hypertree, sometimes called an arboreal hypergraph, if there exists a (non-hyper) tree such that the nodes of each hyperedge of the hypertree correspond to a subtree of the tree. Naturally then, a hyperforest is a hypergraph comprised of one or more disjoint hypertrees.
Two hypergraphs are isomorphic if there exists a bijection between the nodes that induces a bijection on the corresponding hyperedges. I.e. isomorphic hypergraphs are structurally the same. What is the number of hyperforests on ten nodes, distinct up to isomorphism, for which no node is isolated?