Combinatorics

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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.

In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a k-combination of a set S is a subset of k distinct elements of S. So, two combinations are identical if and only if each combination has the same members. (The arrangement of the members in each set does not mat

German Jesuit scholar (1601 or 1602-1680)

mathematical lemma that, if 𝑛 items are put into 𝑚 containers, with 𝑛>𝑚, then at least one container must contain more than one item

family of positive integers that occur as coefficients in the binomial theorem

mathematical ways to group elements of a set

sequence or array in which each further term is defined as a function of the preceding terms

thumb|In coffee percolation, soluble compounds leave the coffee grounds and join the water to form coffee. Insoluble compounds (and granulates) remain within the [[coffee filter.]] thumb|Percolation in a square lattice.

chemical methods designed to rapidly synthesize large numbers of chemical compounds

value of the Dirichlet beta function with two as the argument

theorem about how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem to polynomials

matrix that shows the relationship between two classes of objects

selection of some objects in a particular order

function that is invariant under all permutations of its variables

combinatorial principle to determine possible ways to match disjoint sets.

theorem

mathematical problem of finding a stable matching between two equally sized sets of elements given an ordering of preferences for each element

area of mathematics

the problem of finding a sequence that is a subsequence of each of a given set of sequences and is as long as possible

mathematical counting-out question

counting principle in combinatorics

mathematical notation

combinatorial identity about binomial coefficients

mathematical method to measure sets of natural numbers

algorithm to find the longest increasing subsequence in an array of numbers

graphical aid for deriving certain combinatorial theorems

rapid growth of the complexity of a problem due to how the combinatorics of the problem is affected by the input, constraints, and bounds of the problem

Characterization of large sets

In mathematics, a '''q-analog' of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as . Typically, mathematicians are interested in q-analogs that arise naturally, rather than in arbitrarily contriving q-analogs of known results. The earliest q''-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.

historical term in mathematics

formula in Lie theory

number of paths between grid corners, allowing diagonal steps

mixed radix numeral system adapted to numbering permutations; represents a number as a×0! + b×1! + c×2! + ⋯

in discrete mathematics, a pair of positions in a sequence where two elements are out of sorted order

topological space with a finite number of points

set together with a family of subsets in math

combinatorial methods used in combinatorics, a branch of mathematics

lemma that every Sperner coloring of a triangulated simplex contains a properly colored simplex

conjecture in combinatorial number theory

an abstract mathematical object generalizing the properties of points and lines in the Euclidean plane

lemma proving that a set of events with limited dependence has positive probability of all happening together

Theorem on the existence of finite sets of positive integers >1 whose inverses sum to 1

a pair of six-sided dice with nonstandard labels whose sums have the same probability distribution as standard dice

collection of sets in which every two sets have the same intersection

musical dice game used to randomly generate music

aspect of history

theorem that every aperiodic strongly-connected out-regular directed graph can be labeled to give a synchronizable deterministic finite automaton

nonconstructive method for mathematical proofs

Geometric construct

classic computing problem

Polynomial in combinatorial mathematics

ternary relation that is cyclic (if [𝑥,𝑦,𝑧] then [𝑧,𝑥,𝑦]), asymmetric (if [𝑥,𝑦,𝑧] then not [𝑧,𝑦,𝑥]), transitive (if [𝑤,𝑥,𝑦] and [𝑤,𝑦,𝑧] then [𝑤,𝑥,𝑧]) and connected (for distinct 𝑥,𝑦,𝑧, either [𝑥,𝑦,𝑧] or [𝑧,𝑥,𝑦])

mathematical subject

self-avoiding closed curve which intersects a line a number of times

a sequence of integers constructed as an arithmetic progression

class of sequences of natural numbers

Concept in mathematical graph theory

relation in algebra

Sequence of integers