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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) ...

Pre-Algebra

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 ...

Let a random n×n (0,1)-matrix have entries which are 1 (with probability p) or 0 (with probability q=1-p). An s-cluster is an isolated group of s adjacent (i.e., horizontally ...

Lachlan's term for a set of four points, no three of which are collinear.

A series-reduced tree is a tree in which all nodes have degree other than 2 (in other words, no node merely allows a single edge to "pass through"). Series-reduced trees are ...

A leaf of an unrooted tree is a node of vertex degree 1. Note that for a rooted or planted tree, the root vertex is generally not considered a leaf node, whereas all other ...

Two lines in two-dimensional Euclidean space are said to be parallel if they do not intersect. In three-dimensional Euclidean space, parallel lines not only fail to ...

An unlabeled transitive digraph with n nodes is called a "topology." The numbers of distinct topologies on n=1, 2, ... nodes are 1, 3, 9, 33, 139, 718, 4535, ... (OEIS ...

A set of n magic circles is a numbering of the intersections of the n circles such that the sum over all intersections is the same constant for all circles. The above sets of ...