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Macaulay Representation Of An Integer
Given positive integers n and d, the d-th Macaulay representation of n is an expression for n as a sum of binomial coefficients: : n = \binom + \binom + \cdots + \binom + \binom. Here, c_1,\ldots,c_d is a uniquely determined, strictly increasing sequence of nonnegative integers known as the Macaulay coefficients. For any two positive integers n_1 and n_2, n_1 is less than n_2 if and only if the sequence of Macaulay coefficients for n_1 comes before the sequence of Macaulay coefficients for n_2 in lexicographic order. Macaulay coefficients are also known as the combinatorial number system In mathematics, and in particular in combinatorics, the combinatorial number system of degree ''k'' (for some positive integer ''k''), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natura .... References * * * {{algebra-stub Factorial and binomial topics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. 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 natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...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 n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Increasing Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infinite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lexicographic Order
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set. There are several variants and generalizations of the lexicographical ordering. One variant applies to sequences of different lengths by comparing the lengths of the sequences before considering their elements. Another variant, widely used in combinatorics, orders subsets of a given finite set by assigning a total order to the finite set, and converting subsets into increasing sequences, to which the lexicographical order is applied. A generalization defines an order on a Cartesian product of partially ordered sets; this order is a total order if and only if all factors of the Cartesian product are totally ordered. Motivation and definition The words in a lexicon (the set of words used in some language) have a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Number System
In mathematics, and in particular in combinatorics, the combinatorial number system of degree ''k'' (for some positive integer ''k''), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natural numbers (taken to include 0) ''N'' and ''k''-combinations. The combinations are represented as strictly decreasing sequences ''c''''k'' > ... > ''c''2 > ''c''1 ≥ 0 where each ''ci'' corresponds to the index of a chosen element in a given ''k''-combination. Distinct numbers correspond to distinct ''k''-combinations, and produce them in lexicographic order. The numbers less than \tbinom nk correspond to all of . The correspondence does not depend on the size ''n'' of the set that the ''k''-combinations are taken from, so it can be interpreted as a map from N to the ''k''-combinations taken from N; in this view the correspondence is a bijection. The number ''N'' corresponding to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CRC Press
The CRC Press, LLC is an American publishing group that specializes in producing technical books. Many of their books relate to engineering, science and mathematics. Their scope also includes books on business, forensics and information technology. CRC Press is now a division of Taylor & Francis, itself a subsidiary of Informa. History The CRC Press was founded as the Chemical Rubber Company (CRC) in 1903 by brothers Arthur, Leo and Emanuel Friedman in Cleveland, Ohio, based on an earlier enterprise by Arthur, who had begun selling rubber laboratory aprons in 1900. The company gradually expanded to include sales of laboratory equipment to chemists. In 1913 the CRC offered a short (116-page) manual called the ''Rubber Handbook'' as an incentive for any purchase of a dozen aprons. Since then the ''Rubber Handbook'' has evolved into the CRC's flagship book, the '' CRC Handbook of Chemistry and Physics''. In 1964, Chemical Rubber decided to focus on its publishing ventures ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Publishing
Springer Publishing Company is an American publishing company of academic journals and books, focusing on the fields of nursing, gerontology, psychology, social work, counseling, public health, and rehabilitation (neuropsychology). It was established in 1951 by Bernhard Springer, a great-grandson of Julius Springer, and is based in Midtown Manhattan, New York City. History Springer Publishing Company was founded in 1950 by Bernhard Springer, the Berlin-born great-grandson of Julius Springer, who founded Springer-Verlag (now Springer Science+Business Media). Springer Publishing's first landmark publications included ''Livestock Health Encyclopedia'' by R. Seiden and the 1952 ''Handbook of Cardiology for Nurses''. The company's books soon branched into other fields, including medicine and psychology. Nursing publications grew rapidly in number, as Modell's ''Drugs in Current Use'', a small annual paperback, sold over 150,000 copies over several editions. Solomon Garb's ''Labor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
