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In combinatorial mathematics, the Prüfer sequence ( also Prüfer code or Prüfer numbers ) of a labeled tree is a unique sequence associated with the tree.
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He joined Bell Laboratories, then decided to continue his education and attended the University of Pennsylvania, and studied combinatorial mathematics.
Garfield studied under Herbert Wilf and earned a Ph. D. in combinatorial mathematics from Penn in 1993.
For example, Mancala is a mathematical game, because mathematicians can study it using combinatorial game theory, even though no mathematics is necessary in order to play it.
In combinatorial mathematics, a Steiner system ( named after Jakob Steiner ) is a type of block design, specifically a t-design with λ = 1 and t ≥ 2.
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element.
* Block design, a kind of set system in combinatorial mathematics
Magic squares were known to Chinese mathematicians, as early as 650 BCE and Arab mathematicians, possibly as early as the 7th century, when the Arabs conquered northwestern parts of the Indian subcontinent and learned Indian mathematics and astronomy, including other aspects of combinatorial mathematics.
Vladimir Batagelj is a Slovenian mathematician, born 1948 in Idrija, Slovenia, who works mainly in data analysis, discrete mathematics, combinatorial optimization and applications of IT in education.
* Incidence structure, a feature of combinatorial mathematics
The assignment problem is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics.
In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for analysing discrete structures in terms of generating functions.
In mathematics, the nimbers, also called Grundy numbers, are introduced in combinatorial game theory, where they are defined as the values of nim heaps.
In combinatorial mathematics, Hall's marriage theorem, or simply Hall's Theorem, gives a necessary and sufficient condition for being able to select a distinct element from each of a collection of finite sets.
In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects.
In mathematics, a building ( also Tits building, Bruhat – Tits building, named after François Bruhat and Jacques Tits ) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces.
During the twentieth century this was a fruitful area of combinatorial mathematics, with numerous connections to other fields.
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces ( for example the Betti numbers ) were regarded as derived from combinatorial decompositions such as simplicial complexes.
In applied mathematics and theoretical computer science, combinatorial optimization
In combinatorial mathematics, the necklace polynomials, or ( Moreau's ) necklace-counting function are the polynomials in α such that
* WZ theory, a technique for simplifying certain combinatorial summations in mathematics
In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which follows from it.
In combinatorial mathematics, an ordered partition O of a set S is a sequence
combinatorial and sequence
Forward differences applied to a sequence are sometimes called the binomial transform of the sequence, and have a number of interesting combinatorial properties.
Multiple sequence alignments are computationally difficult to produce and most formulations of the problem lead to NP-complete combinatorial optimization problems.
In combinatorics, a combinatorial class ( or simply class ) is an equivalence class of sets that have the same counting sequence.
In combinatorial mathematics, a k-ary De Bruijn sequence B ( k, n ) of order n, named after the Dutch mathematician Nicolaas Govert de Bruijn, is a cyclic sequence of a given alphabet A with size k for which every possible subsequence of length n in A appears as a sequence of consecutive characters exactly once.
In combinatorial mathematics, the Stirling transform of a sequence
combinatorial and also
New types of Design of experiments methods have also been developed to efficiently address the large experimental spaces that can be tackled using combinatorial methods.
Since the early 1960s, with the availability of oracles for certain combinatorial games, also called tablebases ( e. g. for 3x3-chess ) with any beginning configuration, small-board dots-and-boxes, small-board-hex, and certain endgames in chess, dots-and-boxes, and hex ; a new area for data mining has been opened.
More formally, a set R X Y is called a ( combinatorial ) rectangle if whenever R and R then R. Equivalently, R can also be viewed as a submatrix of the input matrix A such that R = M N where M X and N Y.
Games like nim also admit of a rigorous analysis using combinatorial game theory.
In digital circuit theory, combinational logic ( sometimes also referred to as combinatorial logic ) is a type of digital logic which is implemented by boolean circuits, where the output is a pure function of the present input only.
Following an idea of Grassmann's father, A1 also defined the exterior product, also called " combinatorial product " ( In German: äußeres Produkt or kombinatorisches Produkt ), the key operation of an algebra now called exterior algebra.
It is also exploited as a reasonably efficient mechanism for performing highly combinatorial evaluations of facts where large numbers of joins must be performed between fact tuples.
Brouwer also proved the simplicial approximation theorem in the foundations of algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes, of the treatment of general continuous mappings.
Affine geometry can be viewed as the geometry of affine space, of a given dimension n, coordinatized over a field K. There is also ( in two dimensions ) a combinatorial generalization of coordinatized affine space, as developed in synthetic finite geometry.
This also illustrates how proof theory can be viewed as operating on proofs in a combinatorial fashion: given proofs for both A and B, one can construct a proof for A ∧ B.
They also include one-player combinatorial puzzles, and even no-player automata, like Conway's Game of Life.
CGT has a different emphasis than " traditional " or " economic " game theory, which was initially developed to study games with simple combinatorial structure, but with elements of chance ( although it also considers sequential moves, see extensive-form game ).
In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set.
His father Derrick Norman Lehmer, known mainly as a pioneer in number theory computing, also made major contributions to combinatorial computing, having devised algorithms for efficiently generating all the permutations on n elements He also developed two algorithms, rank ( p ) and unrank ( k ).
* Selection of which studies to include in a meta-analysis ( see also combinatorial meta-analysis ).
Babbitt also described the semi-combinatorial row and the all-combinatorial row, the latter being a row which is combinatorial with any of its derivations and their transpositions.
His own use of it was inimitable ; in terms of later techniques, it is recognised as a prototype of the large sieve method in its application of bilinear forms, and also as an exploitation of combinatorial structure.
The book also contained results in linear diophnatine analysis, decimal periods, combinations, and gave combinatorial significance to the digits of numbers written in decimal notation.
See also combinatorial optimization problems.
The term " combinatorial proof " may also be used more broadly to refer to any kind of elementary proof in combinatorics.
Additionally, multidimensional combinatorial problems, including most design problems in engineering such as form-finding and behavior-finding, suffer from the curse of dimensionality, which also makes them infeasible for exhaustive search or analytical methods.
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