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Systematic classification of 12 related enumerative problems concerning two finite sets
In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical
Twelvefold_way
Topics referred to by the same term
several articles. Described together, in-depth: Twelvefold way Explained separately in a more accessible way: Combination Permutation For meanings outside
Combinations_and_permutations
Probabilities of winning a lottery game
lottery game. It is based primarily on combinatorics, particularly the twelvefold way and combinations without replacement. It can also be used to analyze
Lottery_mathematics
Topics referred to by the same term
distributions of m objects to n recipients, subject to various conditions; see Twelvefold way Distribution (concurrency), the projection operator in a history monoid
Distribution
Graphical aid for deriving some concepts in combinatorics
{n-1}{k-1}}} . Gaussian binomial coefficient Partition (number theory) Twelvefold way Dirichlet-multinomial distribution Batterson, J. Competition Math for
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
1713 book on probability and combinatorics by Jacob Bernoulli
subject. It also addressed problems that today are classified in the twelvefold way and added to the subjects; consequently, it has been dubbed an important
Ars_Conjectandi
Branch of discrete mathematics
is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a unified framework for counting permutations, combinations
Combinatorics
Area of combinatorics that deals with the number of ways certain patterns can be formed
applications have a relatively simple combinatorial description. The twelvefold way provides a unified framework for counting permutations, combinations
Enumerative_combinatorics
Study of discrete mathematical structures
on counting the number of certain combinatorial objects – e.g. the twelvefold way provides a unified framework for counting permutations, combinations
Discrete_mathematics
Mathematical theorem on convolved binomial coefficients
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Vandermonde's_identity
Number theory expression
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Legendre's_formula
Sequence of numbers ((2n) choose (n))
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Central_binomial_coefficient
Recurrence relations of binomial coefficients in Pascal's triangle
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Hockey-stick_identity
Type of permutation of a set of elements
besides Pi, the unreceivable hat is hj, while for Pi it is h1). Another way to see this is to rename h1 to hi, where the derangement is more explicit:
Derangement
Mathematical function such that every output has at least one input
The cardinality of this set is one of the twelve aspects of Rota's Twelvefold way, and is given by | B | ! { | A | | B | } {\textstyle |B|
Surjective_function
Mathematical function
only if x = 1 {\displaystyle x=1} . The asymptotic expansion gives an easy way to compute ψ(x) when the real part of x is large. To compute ψ(x) for small
Digamma_function
Number computed as a product of powers
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Hyperfactorial
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Alternating_factorial
Generalization of the mathematical factorial
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Bhargava_factorial
Conjecture in combinatorial number theory
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Singmaster's_conjecture
Decomposition of an integer as a sum of positive integers
notion of rank Smallest-parts function Stars and bars (combinatorics) Twelvefold way Andrews 1976, p. 199. Josuat-Vergès, Matthieu (2010), "Bijections between
Integer_partition
Mathematical series
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Binomial_series
Recursive integer sequence
The number of reflected paths can be counted in the usual way, by counting how many way upward steps may be distributed among total steps, which is
Catalan_number
Italian-American mathematician (1932–1999)
introduced by Rota in the early 1960s Cyclotomic identity Necklace ring Twelvefold way List of American philosophers O'Connor, John J.; Robertson, Edmund F
Gian-Carlo_Rota
Theorem on prime numbers
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Wilson's_theorem
Probability distribution
undesirable. In such cases there are various alternative estimators. One way is to use the Bayes estimator p ^ b {\displaystyle {\widehat {p}}_{b}} ,
Binomial_distribution
Algebraic expansion of powers of a binomial
binomial. However, there will be several terms of the form xn−2y2, one for each way of choosing exactly two binomials to contribute a y. Therefore, after combining
Binomial_theorem
Index of articles associated with the same name
in the Josephus permutation Ordered selections and partitions of the twelvefold way in combinatorics Ordered set, a bijection, cyclic order, or permutation
Order_(mathematics)
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Stirling_transform
Stirling transform Stratification (mathematics) Tverberg partition Twelvefold way Chinese restaurant process Dobinski's formula Ewens's sampling formula
List_of_partition_topics
Generalization of the binomial theorem to other polynomials
triangle or Pascal's pyramid to Pascal's simplex. This provides a quick way to generate a lookup table for multinomial coefficients. A related structure
Multinomial_theorem
Transformation of a mathematical sequence
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Binomial_transform
Triangular array of the binomial coefficients
84}}\quad 280\quad 560\quad 672\quad 448\quad 128\end{matrix}}} The other way of producing this triangle is to start with Pascal's triangle and multiply
Pascal's_triangle
Mathematical version of an order change
numbers Sorting network Substitution cipher Superpattern Superpermutation Twelvefold way Weak order of permutations 1 is frequently used to represent the identity
Permutation
Uniqueness theorem in complex analysis
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Carlson's_theorem
Formula in mathematics
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Trinomial_expansion
Recursive mathematical formula
n} defined by iterating the process of exponentiation, analogously to the way that the factorial can be defined by iterated multiplication. The exponential
Exponential_factorial
Number of subsets of a given size
has an integer value at all integer inputs t {\displaystyle t} . (One way to prove this is by induction on k using Pascal's identity.) Therefore, any
Binomial_coefficient
Numbers parameterizing ways to partition a set
the number of partitions of a set with n members Stirling polynomials Twelvefold way Ronald L. Graham, Donald E. Knuth, Oren Patashnik (1988) Concrete Mathematics
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Arithmetical function
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Kempner_function
Product of consecutive factorial numbers
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Superfactorial
Probability distribution
property persists when the definition is thus generalized, and affords a quick way to see that the negative binomial distribution is infinitely divisible. The
Negative binomial distribution
Negative_binomial_distribution
Numerical method for calculating the gamma function
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Lanczos_approximation
Overview of and topical guide to combinatorics
combinatorics Symbolic combinatorics Combinatorial class Exponential formula Twelvefold way MacMahon Master theorem Data structure Data type Abstract data type
Outline_of_combinatorics
Theorem in p-adic analysis
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Mahler's_theorem
Generalization of the binomial distribution
is sometimes used for the categorical distribution to emphasize this four-way relationship (so n determines the suffix, and k the prefix). The Bernoulli
Multinomial_distribution
Solved prime-number problem
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Proof_of_Bertrand's_postulate
Mathematical result on arithmetic properties of binomial coefficients
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Star_of_David_theorem
Numbering of combinations of items
contained in {0, 1, ..., n − 1}, the k-combinations must be ordered in such a way that their largest elements are compared first. The most natural ordering
Combinatorial_number_system
Mathematical sequences in combinatorics
coefficients in any basis are unique, one can define Stirling numbers this way, as the coefficients expressing polynomials of one basis in terms of another
Stirling_number
Mathematical functions
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Falling_and_rising_factorials
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Difference_polynomials
Product of the first "n" prime numbers
"primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors. The primorial p n # {\displaystyle
Primorial
Arrangement of trinomial coefficients
they are not individually confused with Pascal's triangle. There is three-way symmetry of the numbers in each layer. The number of terms in the nth layer
Pascal's_pyramid
Type of prime number
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Wilson_prime
Fundamental Buddhist teaching
separate links in the twelvefold list, so this equation can't be correct for this nidana. According to Erich Frauwallner, the twelvefold chain resulted from
Pratītyasamutpāda
Expectation or average of the falling factorial of a random variable
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Factorial_moment
Triangular array of natural numbers
( n , 1 ) = 1 {\displaystyle \operatorname {N} (n,1)=1} , since the only way to get a single sub-pattern () is to have all the opening parentheses in
Narayana_number
Numeral system in combinatorics
that can be converted to a permutation of n elements in a straightforward way, either using them as Lehmer code or as inversion table representation; in
Factorial_number_system
permutations Stanley–Wilf conjecture Symmetric function Szymanski's conjecture Twelvefold way Alternating group Automorphisms of the symmetric and alternating groups
List_of_permutation_topics
Upper bound on intersecting set families
and are all subsets of some larger set of size n {\displaystyle n} . One way to construct a family of sets with these parameters, each two sharing an
Erdős–Ko–Rado_theorem
Mathematical integral
roughly, it can be said to be related to the Nörlund–Rice integral in the same way that Perron's formula is related to the Mellin transform: rather than dealing
Nørlund–Rice_integral
Family of polynomials
q 4 {\displaystyle {4 \choose 2}_{\!q}=1+q+2q^{2}+q^{3}+q^{4}} . Another way to see this is to associate each word with a path across a rectangular grid
Gaussian_binomial_coefficient
Integer sequence
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Gould's_sequence
Mathematics term
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Lozanić's_triangle
Type of number in combinatorial mathematics and statistics
allowed ways of arranging a number of articles, that is restricted in some way. This means that they are related to the binomial coefficient. The key difference
Fuss–Catalan_number
Type of polynomial sequence
polynomial sequence of binomial type is determined by its cumulants, in a way discussed in the article titled cumulant. Thus p n ′ ( 0 ) = κ n = {\displaystyle
Binomial_type
Generalized chain rule in calculus
m_{n}=n.} Sometimes, to give it a memorable pattern, it is written in a way in which the coefficients that have the combinatorial interpretation discussed
Faà_di_Bruno's_formula
Mathematical identity involving sums of binomial coefficients
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Abel's_binomial_theorem
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Binomial_QMF
Infinite matrices with Pascal's triangle as elements
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Pascal_matrix
result is: V R 3 , 8 = 3 8 = 6561 {\displaystyle VR_{3,8}=3^{8}=6561} Twelvefold way Dubois, Jean-Guy. Une systématique des configurations combinatoires
Combinatorial_modelling
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Trinomial_triangle
Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion · Pigeonhole principle
Egorychev_method
Mathematical sequence
\{3,(2,1)\}} L ( n , n ) {\textstyle L(n,n)} is always 1 because the only way to partition { 1 , 2 , … , n } {\textstyle \{1,2,\ldots ,n\}} into n {\displaystyle
Lah_number
Buddhist canonical collection
which contained texts that were later termed "Mahayana". The Dvādaśāṅga ("twelvefold [scriptural] division") refers to a traditional Indian classification
Buddhist_canons
Number that is the product of factorials
that is a Jordan–Pólya number, and every Jordan–Pólya number arises in this way as the order of an automorphism group of a tree. These numbers are named
Jordan–Pólya_number
Buddhist staff
rings representing the Six Perfections, or twelve rings representing the twelvefold chain of cause and effect. A notable carrier of the staff is Kṣitigarbha
Khakkhara
American non-profit parapsychological research institute
after its publication "traffic to [the institute's] website ... increased twelvefold", applications for membership increased and "journalists from places like
Institute_of_Noetic_Sciences
Creator god in Hinduism
Vishnu. Verily, that One became threefold, became eightfold, elevenfold, twelvefold, into infinite fold. This Being (neuter) entered all beings, he became
Brahma
Subdivision of signs into 12 sub parts
of a point in the heavens is computed by adding the degree multiplied twelvefold not to 0° of its sign but to the exact degree in question. In symbolic
Dodecatemoria
Set of all pitches that are a whole number of octaves apart
it eliminates any suggestion that there is something natural about a twelvefold division of the octave. Second, it avoids pitch-class universes with unwieldy
Pitch_class
Pure concept of the understanding in Kantianism
by Kant as a model for the table of categories. Taken together, these twelvefold tables constitute the formal structure for Kant's architectonic conception
Category_(Kant)
System of Renaissance magic
Hexagram, a surrounding border of Enochian letters, and in the middle a Twelvefold table (cell) engraved with individual Enochian letters. According to Duquette
Enochian_magic
President of Iraq from 1968 to 1979
became a net importer of food grains; imports of food grain increased twelvefold from the early 1960s. The introduction of subsidies and the removal of
Ahmed_Hassan_al-Bakr
English astrologer (1920–1982)
constructs such as the twelvefold zodiac and house systems were unable to evaluate many of the effects of cosmic periods. Twelvefold systems allow patterns
John_Addey_(astrologer)
Early Buddhist school
original teachings of the Buddha had been lost. The Dharmaguptaka used a twelvefold division of the Buddhist teachings, which has been found in their Dīrgha
Dharmaguptaka
Sutra in Mahāyāna Buddhism
Mahāvastu and the Mūlasarvāstivādin Vinaya, as the Sanskrit parallel to the twelvefold Pali formulation inscribed on Dvaravati-period wheels of the law in Thailand
Lalitavistara_Sūtra
Irish businessman and rugby union player (1936–2024)
the company into a major international competitor, its value increasing twelvefold (from $908 million to $11 billion). O'Reilly left Heinz in 1998 after
Tony_O'Reilly
Task of finding records in a data set that refer to same entity across different sources
stable but creates only twelve groups, cutting comparisons only about twelvefold, which for very large files may still leave too many pairs. Systems therefore
Record_linkage
Five aggregates of clinging in Buddhism
skandhas. Eventually, this branched list developed into the standard twelvefold chain as a linear list. According to Boisvert, "the function of each of
Skandha
Chinese Buddhist master who founded the Tiantai school
mundane or conventional truth in Madhyamaka, which Zhiyi glosses as "the twelvefold conditioned co-arising of ignorance" and as "illusory existence". In Zhiyi's
Zhiyi
Highest spiritual goal in Buddhist cultivation
insight of the Buddha eventually were superseded by Pratītyasamutpāda, the twelvefold chain of causation, and still later by anatta, the emptiness of the self
Enlightenment_in_Buddhism
English poet, novelist and librarian (1922–1985)
sextupled, and the budget expanded from £4,500 to £448,500, in real terms a twelvefold increase. Dockery, now: Only nineteen, he must have taken stock Of what
Philip_Larkin
Methane-water lattice compound
believe the methane content of the planet's atmosphere would increase twelvefold. Preuss, Paul (17 September 2008). "IMPACTS: On the Threshold of Abrupt
Methane_clathrate
communication.... Jehovah's visible organization is based firmly on the twelvefold foundation of the apostles of the Lamb with Jesus Christ himself being
Criticism of Jehovah's Witnesses
Criticism_of_Jehovah's_Witnesses
Television station in Elkhart, Indiana
and postponed the planned March 17 sign-on date. Power was increased twelvefold in November when a new transmitter was installed. The station originally
WSJV
Action driven by intention which leads to future consequences
The twelvefold chain as we know it is the result of a gradual development. Shorter versions are also known. According to Schumann, the twelvefold chain
Karma_in_Buddhism
nearly US$160 billion in 1980 to over US$2 trillion by 1991, a more than twelvefold increase. Liabilities—investments by foreigners in Japan—expanded somewhat
Capital_flows_in_Japan
Chapter of the Hebrew and Christian Bible
twelve actions by Josiah, which in numerological view is signified by his 'twelvefold purging' of idolatry, reformation of all twelve tribes of Israel and the
2_Kings_23
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