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Twelvefold Way
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. The idea of the classification is credited to Gian-Carlo Rota, and the name was suggested by Joel Spencer. Overview Let and be finite sets. Let n=, N, and x=, X, be the cardinality of the sets. Thus is an -set, and is an -set. The general problem we consider is the enumeration of equivalence classes of functions f: N \to X. The functions are subject to one of the three following restrictions: # No condition: each in may be sent by to any in , and each may occur multiple times. # is injective: each value f(a) for in must be distinct from every other, and so each in may occur at most once in the image of . # is surjective: for each in there must be at least one in such that f(a) = b, thus e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in 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. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Case Sn
Case or CASE may refer to: Containers * Case (goods), a package of related merchandise * Cartridge case or casing, a firearm cartridge component * Bookcase, a piece of furniture used to store books * Briefcase or attaché case, a narrow box to carry paperwork * Computer case, the enclosure for a PC's main components * Keep case, DVD or CD packaging * Pencil case * Phone case, protective or vanity accessory for mobile phones ** Battery case * Road case or flight case, for fragile equipment in transit * Shipping container or packing case * Suitcase, a large luggage box * Type case, a compartmentalized wooden box for letterpress typesetting Places * Case, Laclede County, Missouri * Case, Warren County, Missouri * Case River, a Kabika tributary in Ontario, Canada * Case Township, Michigan * Case del Conte, Italy People * Case (name), people with the surname (or given name) * Case (singer), American R&B singer-songwriter and producer (Case Woodard) Arts, entertainment, and med ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composition (number Theory)
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 partition of that number. Every integer has finitely many distinct compositions. Negative numbers do not have any compositions, but 0 has one composition, the empty sequence. Each positive integer ''n'' has distinct compositions. A weak composition of an integer ''n'' is similar to a composition of ''n'', but allowing terms of the sequence to be zero: it is a way of writing ''n'' as the sum of a sequence of non-negative integers. As a consequence every positive integer admits infinitely many weak compositions (if their length is not bounded). Adding a number of terms 0 to the ''end'' of a weak composition is usually not considered to define a different weak composition; in other words, weak compositions a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Case Sx
Case or CASE may refer to: Containers * Case (goods), a package of related merchandise * Cartridge case or casing, a firearm cartridge component * Bookcase, a piece of furniture used to store books * Briefcase or attaché case, a narrow box to carry paperwork * Computer case, the enclosure for a PC's main components * Keep case, DVD or CD packaging * Pencil case * Phone case, protective or vanity accessory for mobile phones ** Battery case * Road case or flight case, for fragile equipment in transit * Shipping container or packing case * Suitcase, a large luggage box * Type case, a compartmentalized wooden box for letterpress typesetting Places * Case, Laclede County, Missouri * Case, Warren County, Missouri * Case River, a Kabika tributary in Ontario, Canada * Case Township, Michigan * Case del Conte, Italy People * Case (name), people with the surname (or given name) * Case (singer), American R&B singer-songwriter and producer (Case Woodard) Arts, entertainment, and medi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Case Fn
Case or CASE may refer to: Containers * Case (goods), a package of related merchandise * Cartridge case or casing, a firearm cartridge component * Bookcase, a piece of furniture used to store books * Briefcase or attaché case, a narrow box to carry paperwork * Computer case, the enclosure for a PC's main components * Keep case, DVD or CD packaging * Pencil case * Phone case, protective or vanity accessory for mobile phones ** Battery case * Road case or flight case, for fragile equipment in transit * Shipping container or packing case * Suitcase, a large luggage box * Type case, a compartmentalized wooden box for letterpress typesetting Places * Case, Laclede County, Missouri * Case, Warren County, Missouri * Case River, a Kabika tributary in Ontario, Canada * Case Township, Michigan * Case del Conte, Italy People * Case (name), people with the surname (or given name) * Case (singer), American R&B singer-songwriter and producer (Case Woodard) Arts, entertainment, and med ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Case S
Case or CASE may refer to: Containers * Case (goods), a package of related merchandise * Cartridge case or casing, a firearm cartridge component * Bookcase, a piece of furniture used to store books * Briefcase or attaché case, a narrow box to carry paperwork * Computer case, the enclosure for a PC's main components * Keep case, DVD or CD packaging * Pencil case * Phone case, protective or vanity accessory for mobile phones ** Battery case * Road case or flight case, for fragile equipment in transit * Shipping container or packing case * Suitcase, a large luggage box * Type case, a compartmentalized wooden box for letterpress typesetting Places * Case, Laclede County, Missouri * Case, Warren County, Missouri * Case River, a Kabika tributary in Ontario, Canada * Case Township, Michigan * Case del Conte, Italy People * Case (name), people with the surname (or given name) * Case (singer), American R&B singer-songwriter and producer (Case Woodard) Arts, entertainment, and med ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Case In
Case or CASE may refer to: Containers * Case (goods), a package of related merchandise * Cartridge case or casing, a firearm cartridge component * Bookcase, a piece of furniture used to store books * Briefcase or attaché case, a narrow box to carry paperwork * Computer case, the enclosure for a PC's main components * Keep case, DVD or CD packaging * Pencil case * Phone case, protective or vanity accessory for mobile phones ** Battery case * Road case or flight case, for fragile equipment in transit * Shipping container or packing case * Suitcase, a large luggage box * Type case, a compartmentalized wooden box for letterpress typesetting Places * Case, Laclede County, Missouri * Case, Warren County, Missouri * Case River, a Kabika tributary in Ontario, Canada * Case Township, Michigan * Case del Conte, Italy People * Case (name), people with the surname (or given name) * Case (singer), American R&B singer-songwriter and producer (Case Woodard) Arts, entertainment, and medi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Case I
Case or CASE may refer to: Containers * Case (goods), a package of related merchandise * Cartridge case or casing, a firearm cartridge component * Bookcase, a piece of furniture used to store books * Briefcase or attaché case, a narrow box to carry paperwork * Computer case, the enclosure for a PC's main components * Keep case, DVD or CD packaging * Pencil case * Phone case, protective or vanity accessory for mobile phones ** Battery case * Road case or flight case, for fragile equipment in transit * Shipping container or packing case * Suitcase, a large luggage box * Type case, a compartmentalized wooden box for letterpress typesetting Places * Case, Laclede County, Missouri * Case, Warren County, Missouri * Case River, a Kabika tributary in Ontario, Canada * Case Township, Michigan * Case del Conte, Italy People * Case (name), people with the surname (or given name) * Case (singer), American R&B singer-songwriter and producer (Case Woodard) Arts, entertainment, and medi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Permutation
In combinatorial mathematics, a partial permutation, or sequence without repetition, on a finite set ''S'' is a bijection between two specified subsets of ''S''. That is, it is defined by two subsets ''U'' and ''V'' of equal size, and a one-to-one mapping from ''U'' to ''V''. Equivalently, it is a partial function on ''S'' that can be extended to a permutation. Representation It is common to consider the case when the set ''S'' is simply the set of the first ''n'' integers. In this case, a partial permutation may be represented by a string of ''n'' symbols, some of which are distinct numbers in the range from 1 to n and the remaining ones of which are a special "hole" symbol ◊. In this formulation, the domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ... ''U'' of the part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition (number Theory)
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, can be partitioned in five distinct ways: : : : : : The order-dependent composition is the same partition as , and the two distinct compositions and represent the same partition as . A summand in a partition is also called a part. The number of partitions of is given by the partition function . So . The notation means that is a partition of . Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in general. Examples The seven partitions of 5 are: * 5 * 4 + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stirling Numbers
In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in a purely algebraic setting in his book ''Methodus differentialis'' (1730). They were rediscovered and given a combinatorial meaning by Masanobu Saka in 1782. Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second kind. Additionally, Lah numbers are sometimes referred to as Stirling numbers of the third kind. Each kind is detailed in its respective article, this one serving as a description of relations between them. A common property of all three kinds is that they describe coefficients relating three different sequences of polynomials that frequently arise in combinatorics. Moreover, all three can be defined as the number of partitions of ''n'' elements into ''k'' non-empty subsets, with different ways of counting orderings within each subset. Notation Seve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
