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Legendre's conjecture asserts that for every n there exists a prime p between n^2 and (n+1)^2 (Hardy and Wright 1979, p. 415; Ribenboim 1996, pp. 397-398). It is one of ...

A k-matching in a graph G is a set of k edges, no two of which have a vertex in common (i.e., an independent edge set of size k). Let Phi_k be the number of k-matchings of ...

The maximum leaf number l(G) of a graph G is the largest number of tree leaves in any of its spanning trees. (The corresponding smallest number of leaves is known as the ...

O_h is the point group of symmetries of the octahedron having order 48 that includes inversion. It is also the symmetry group of the cube, cuboctahedron, and truncated ...

Define a pebbling move as a transer of two pebbles from one vertex of a graph edge to an adjacent vertex with one of the pebbles being removed in transit as a toll. The ...

A very general theorem that allows the number of discrete combinatorial objects of a given type to be enumerated (counted) as a function of their "order." The most common ...

To define a recurring digital invariant of order k, compute the sum of the kth powers of the digits of a number n. If this number n^' is equal to the original number n, then ...

A rolling polyhedron graph is a graph obtained by rolling a polyhedral solid along a board whose tiles match up with the faces of the polyhedron being rolled. The vertices of ...

Finch (2001, 2003) defines a k-rough (or k-jagged) number to be positive integer all of whose prime factors are greater than or equal to k. Greene and Knuth define "unusual ...

A sequential substitution system is a substitution system in which a string is scanned from left to right for the first occurrence of the first rule pattern. If the pattern ...