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An Abelian planar difference set of order n exists only for n a prime power. Gordon (1994) has verified it to be true for n<2000000.

sum_(k=0)^dr_k^B(d-k)!x^k=sum_(k=0)^d(-1)^kr_k^(B^_)(d-k)!x^k(x+1)^(d-k).

As shown by Schur (1916), the Schur number S(n) satisfies S(n)<=R(n)-2 for n=1, 2, ..., where R(n) is a Ramsey number.

A tournament sequence is an increasing sequence of positive integers (t_1, t_2, ...) such that t_1=1 and t_(i+1)<=2t_i. Cook and Kleber (2000) show that Meeussen sequences ...

The Fibonacci number F_(n+1) gives the number of ways for 2×1 dominoes to cover a 2×n checkerboard, as illustrated in the diagrams above (Dickau). The numbers of domino ...

The Fano plane is the configuration consisting of the two-dimensional finite projective plane over GF(2) ("of order two"), illustrated above. It is a block design with nu=7, ...

In general, an arrangement of objects is simply a grouping of them. The number of "arrangements" of n items is given either by a combination (order is ignored) or permutation ...

Let sigma_1, ..., sigma_4 be four planes in general position through a point P and let P_(ij) be a point on the line sigma_i·sigma_j. Let sigma_(ijk) denote the plane ...

The n-cyclohedron, also known as the Bott-Taubes polytope, is defined as the compactification of the configuration space of n points on the circle. The cyclohedron can be ...

A diminished polyhedron is a polyhedron in which one or more groups of associated faces (such as those comprising the top of a pyramid or cupola in the case of a Johnson ...