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A tree with its nodes labeled. The number of labeled trees on n nodes is n^(n-2), the first few values of which are 1, 1, 3, 16, 125, 1296, ... (OEIS A000272). Cayley (1889) ...

The number of empty neighbors of a cluster.

A sum over all cluster perimeters.

Define a valid "coloring" to occur when no two faces with a common edge share the same color. Given two colors, there is a single way to color an octahedron (Ball and Coxeter ...

The Johnson solids are the convex polyhedra having regular faces and equal edge lengths (with the exception of the completely regular Platonic solids, the "semiregular" ...

The regular dodecahedron, often simply called "the" dodecahedron, is the Platonic solid composed of 20 polyhedron vertices, 30 polyhedron edges, and 12 pentagonal faces, ...

A tree which is not rooted, i.e., a normal tree with no node singled out for special treatment (Skiena 1990, p. 107). Free trees are sometimes known instead as unrooted trees ...

A Hasse diagram is a graphical rendering of a partially ordered set displayed via the cover relation of the partially ordered set with an implied upward orientation. A point ...

While many computations admit shortcuts that allow them to be performed more rapidly, others cannot be sped up. Computations that cannot be sped up by means of any shortcut ...

A nonassociative algebra obeyed by objects such as the Lie bracket and Poisson bracket. Elements f, g, and h of a Lie algebra satisfy [f,f]=0 (1) [f+g,h]=[f,h]+[g,h], (2) and ...