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Composition (combinatorics)
In mathematics, a composition of an integer ''n'' is a way of writing ''n'' as the summation, 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 integer 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 2''n''−1 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partitions Of 6
Partition may refer to: Arts and entertainment Film and television * ''Partition'' (1987 film), directed by Ken McMullen * ''Partition'' (2007 film), directed by Vic Sarin * '' Partition: 1947'', or ''Viceroy's House'', a 2017 film Music * Partition (music), a segment created from a set * "Partition" (song), by Beyoncé, 2014 * ''Partition'', soundtrack to the 2007 film by Brian Tyler Law and politics * Partition (law), the division of an estate * Partition (politics), a change of political borders, including: ** Partition of India ** Partition of Ireland ** Partitions of Poland Science and technology Computing * Disk partitioning, the division of a hard disk drive * Memory partition, a memory management technique * Partition (database), the division of a logical database * Logical partition, a subset of a computer's resources, virtualized as a separate computer * Binary space partitioning, in computer science * Partition problem, in number theory and computer science ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial x^3 + 3 x^2 y + z^7 is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial.However, as some authors do not make a clear distinction between a polynomial and its associated function, the terms ''homogeneous polynomial'' and ''form'' are sometimes considered as synonymous. A binary form is a form in two variables. A ''form'' is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis. A polynomial of degree 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Integer Sequences
The ''Journal of Integer Sequences'' is a peer-reviewed open-access academic journal in mathematics, specializing in research papers about integer sequences. It was founded in 1998 by Neil Sloane. Sloane had previously published two books on integer sequences, and in 1996 he founded the On-Line Encyclopedia of Integer Sequences (OEIS). Needing an outlet for research papers concerning the sequences he was collecting in the OEIS, he founded the journal. Since 2002 the journal has been hosted by the David R. Cheriton School of Computer Science at the University of Waterloo, with Waterloo professor Jeffrey Shallit as its editor-in-chief. There are no page charges for authors, and all papers are free to all readers. The journal publishes approximately 50–75 papers annually.. In most years from 1999 to 2014, SCImago Journal Rank has ranked the ''Journal of Integer Sequences'' as a third-quartile journal in discrete mathematics and combinatorics. It is indexed by ''Mathematical Review ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coefficient
In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless quantity, number without units, in which case it is known as a numerical factor. It may also be a constant (mathematics), constant with units of measurement, in which it is known as a constant multiplier. In general, coefficients may be any mathematical expression, expression (including Variable (mathematics), variables such as , and ). When the combination of variables and constants is not necessarily involved in a product (mathematics), product, it may be called a ''parameter''. For example, the polynomial 2x^2-x+3 has coefficients 2, −1, and 3, and the powers of the variable x in the polynomial ax^2+bx+c have coefficient parameters a, b, and c. A , also known as constant term or simply constant, is a quantity either implicitly attach ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula : \binom nk = \frac, which using factorial notation can be compactly expressed as : \binom = \frac. For example, the fourth power of is : \begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation : \binom = \binom + \binom . The binomial coefficients occur in many areas of mathematics, and espe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Climbing Stairs
Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci'', is first found in a modern source in a 1838 text by the Franco-Italian mathematician Guglielmo Libri and is short for ('son of Bonacci'). However, even as early as 1506, Perizolo, a notary of the Holy Roman Empire, mentions him as "Lionardo Fibonacci". Fibonacci popularized the Indo–Arabic numeral system in the Western world primarily through his composition in 1202 of (''Book of Calculation'') and also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in . Biography Fibonacci was born around 1170 to Guglielmo, an Italian merchant and customs official who directed a trading post in Bugia, modern-day Béjaïa, Algeria. Fibonacci travelled with him as a young boy, and it was in Bugia where he was educat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal Triangle Compositions
Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, French mathematician, physicist, inventor, philosopher, writer and theologian Places * Pascal (crater), a lunar crater * Pascal Island (Antarctica) * Pascal Island (Western Australia) Science and technology * Pascal (unit), the SI unit of pressure * Pascal (programming language), a programming language developed by Niklaus Wirth ** Microsoft Pascal **Turbo Pascal * PASCAL (database), a bibliographic database maintained by the Institute of Scientific and Technical Information * Pascal (microarchitecture), codename for a microarchitecture developed by Nvidia Other uses * (1895–1911) * (1931–1942) * Pascal and Maximus, fictional characters in ''Tangled'' * Pascal blanc, a French white wine grape * Pascal College, secondary education sch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compositions Of 6
Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography * Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include visuals and digital space *Composition (visual arts), the plan, placement or arrangement of the elements of art in a work * ''Composition'' (Peeters), a 1921 painting by Jozef Peeters *Composition studies, the professional field of writing instruction * ''Compositions'' (album), an album by Anita Baker *Digital compositing, the practice of digitally piecing together a still image or video *Musical composition, an original piece of music, or the process of creating a new piece Computer science *Compose key, a key on a computer keyboard *Compositing window manager a component of a computer's graphical user interface that draws windows and/or their borders *Function composition (computer science), an act or mechanism to combine simple funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-negative Integer
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like jersey numbers on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
