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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 ...

A permutation problem invented by Cayley. Let the numbers 1, 2, ..., n be written on a set of cards, and shuffle this deck of cards. Now, start counting using the top card. ...

The odd divisor function sigma_k^((o))(n)=sum_(d|n; d odd)d^k (1) is the sum of kth powers of the odd divisors of a number n. It is the analog of the divisor function for odd ...

A polyhedral nonhamiltonian graph is a graph that is simultaneously polyhedral and nonhamiltonian. The smallest possible number of vertices a nonhamiltonian polyhedral graph ...

A circle having a given number of lattice points on its circumference. The Schinzel circle having n lattice points is given by the equation {(x-1/2)^2+y^2=1/45^(k-1) for n=2k ...

The Sierpiński carpet is the fractal illustrated above which may be constructed analogously to the Sierpiński sieve, but using squares instead of triangles. It can be ...

A planar graph G is said to be triangulated (also called maximal planar) if the addition of any edge to G results in a nonplanar graph. If the special cases of the triangle ...

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 ...

The double-struck capital letter I, I, is a symbol sometimes used instead of Z for the ring of integers. In contrast, the lower case symbol i is used to refer to the ...

An integer that is either 0 or positive, i.e., a member of the set Z^*={0} union Z^+, where Z-+ denotes the positive integers.