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Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures.It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.
PDFtk (short for PDF Toolkit) is a toolkit for manipulating Portable Document Format (PDF) documents. [3] [4] It runs on Linux, Windows and macOS. [5] It comes in three versions: PDFtk Server (open-source command-line tool), PDFtk Free and PDFtk Pro (proprietary paid). [2] It is able to concatenate, shuffle, split and rotate PDF files.
On May 2, 2017, Microsoft unveiled Windows 10 S (referred to in leaks as Windows 10 Cloud), a feature-limited edition of Windows 10 which was designed primarily for devices in the education market (competing, in particular, with ChromeOS netbooks), such as the Surface Laptop that Microsoft also unveiled at this time. The OS restricts software ...
The Fano matroid, derived from the Fano plane.Matroids are one of many kinds of objects studied in algebraic combinatorics. Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
For any pair of positive integers n and k, the number of k-tuples of non-negative integers whose sum is n is equal to the number of multisets of size k − 1 taken from a set of size n + 1, or equivalently, the number of multisets of size n taken from a set of size k, and is given by
Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics.Extremal combinatorics studies how large or how small a collection of finite objects (numbers, graphs, vectors, sets, etc.) can be, if it has to satisfy certain restrictions.
Geometric combinatorics is a branch of mathematics in general and combinatorics in particular. It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn has many applications to computational geometry.
In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of (strictly) positive integers.Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same integer partition of that number.