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In combinatorial mathematics, the series-parallel networks problem asks for the number of networks that can be formed using a given number of edges. The edges can be ...

Dissect a triangle into smaller triangles, such that all have full edge contact with their neighbors. Label the corners 1, 2, and 3. Label all vertices with 1, 2, or 3, with ...

The average number of regions N(n) into which n lines divide a square is N^_(n)=1/(16)n(n-1)pi+n+1 (Santaló 1976; Finch 2003, p. 481). The maximum number of sequences is ...

Let b(k) be the number of 1s in the binary expression of k, i.e., the binary digit count of 1, giving 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, ... (OEIS A000120) for k=1, 2, .... ...

A cubic map is three-colorable iff each interior region is bounded by an even number of regions. A non-cubic map bounded by an even number of regions is not necessarily ...

A vertex coloring is an assignment of labels or colors to each vertex of a graph such that no edge connects two identically colored vertices. The most common type of vertex ...

De Grey (2018) found the first examples of unit-distance graphs with chromatic number 5, thus demonstrating that the solution to the Hadwiger-Nelson problem (i.e., the ...

A k-subset is a subset of a set on n elements containing exactly k elements. The number of k-subsets on n elements is therefore given by the binomial coefficient (n; k). For ...

Given a function f(x)=f_0(x), write f_1=f^'(x) and define the Sturm functions by f_n(x)=-{f_(n-2)(x)-f_(n-1)(x)[(f_(n-2)(x))/(f_(n-1)(x))]}, (1) where [P(x)/Q(x)] is a ...

A quadrilateral, sometimes also known as a tetragon or quadrangle (Johnson 1929, p. 61) is a four-sided polygon. If not explicitly stated, all four polygon vertices are ...