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A solution to Kirkman's schoolgirl problem with vertices denoting girls and colours denoting days of the week [1]. Kirkman's schoolgirl problem is a problem in combinatorics proposed by Thomas Penyngton Kirkman in 1850 as Query VI in The Lady's and Gentleman's Diary (pg.48).
In mathematics, the polynomial method is an algebraic approach to combinatorics problems that involves capturing some combinatorial structure using polynomials and proceeding to argue about their algebraic properties. Recently, the polynomial method has led to the development of remarkably simple solutions to several long-standing open problems ...
In combinatorics, stars and bars (also called "sticks and stones", [1] "balls and bars", [2] and "dots and dividers" [3]) is a graphical aid for deriving certain combinatorial theorems. It can be used to solve a variety of counting problems, such as how many ways there are to put n indistinguishable balls into k distinguishable bins. [4]
[7] [8] With slight alterations this form was adopted by Philippe Flajolet, Robert Sedgewick and Richard P. Stanley in their textbooks on combinatorics. [1] [3] The problem, or riddle, along with a detailed explanation of the solution, was featured by the channel Veritasium in a 2023 on YouTube.
A minimum spanning tree of a weighted planar graph.Finding a minimum spanning tree is a common problem involving combinatorial optimization. Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, [1] where the set of feasible solutions is discrete or can be reduced to a discrete set.
This area is one of the oldest parts of combinatorics, such as in Kirkman's schoolgirl problem proposed in 1850. The solution of the problem is a special case of a Steiner system, which play an important role in the classification of finite simple groups. The area has further connections to coding theory and geometric combinatorics.
Necklace splitting is a picturesque name given to several related problems in combinatorics and measure theory. Its name and solutions are due to mathematicians Noga Alon [1] and Douglas B. West. [2] The basic setting involves a necklace with beads of different colors. The necklace should be divided between several partners (e.g. thieves), such ...
In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number.