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Solid Partition
In mathematics, solid partitions are natural generalizations of partitions and plane partitions defined by Percy Alexander MacMahon. A solid partition of n is a three-dimensional array of non-negative integers n_ (with indices i, j, k\geq 1) such that : \sum_ n_=n and : n_ \leq n_,\quad n_ \leq n_\quad\text\quad n_ \leq n_ for all i, j \text k. Let p_3(n) denote the number of solid partitions of n. As the definition of solid partitions involves three-dimensional arrays of numbers, they are also called three-dimensional partitions in notation where plane partitions are two-dimensional partitions and partitions are one-dimensional partitions. Solid partitions and their higher-dimensional generalizations are discussed in the book by Andrews. Ferrers diagrams for solid partitions Another representation for solid partitions is in the form of Ferrers diagrams. The Ferrers diagram of a solid partition of n is a collection of n points or ''nodes'', \lambda=(\mathbf_1,\mathbf_2,\ld ... [...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 + 1 * 3 + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Partition
In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers \pi_ (with positive number, positive integer indices ''i'' and ''j'') that is nonincreasing in both indices. This means that : \pi_ \ge \pi_ and \pi_ \ge \pi_ for all ''i'' and ''j''. Moreover, only finitely many of the \pi_ may be nonzero. Plane partitions are a generalization of Partition (number theory), partitions of an integer. A plane partition may be represented visually by the placement of a stack of \pi_ unit cubes above the point (''i'', ''j'') in the plane, giving a three-dimensional solid as shown in the picture. The image has matrix form : \begin 4 & 4 & 3 & 2 & 1\\ 4 & 3 & 1 & 1\\ 3 & 2 & 1\\ 1 \end Plane partitions are also often described by the positions of the unit cubes. From this point of view, a plane partition can be defined as a finite subset \mathcal of positive integer lattice points (''i'', ''j'', ''k'') in \mathbb^3, such that if (''r'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Percy Alexander MacMahon
Percy Alexander MacMahon (26 September 1854 – 25 December 1929) was a mathematician, especially noted in connection with the partitions of numbers and enumerative combinatorics. Early life Percy MacMahon was born in Malta to a British military family. His father was a colonel at the time, retired in the rank of the brigadier. MacMahon attended the Proprietary School in Cheltenham. At the age of 14 he won a Junior Scholarship to Cheltenham College, which he attended as a day boy from 10 February 1868 until December 1870. At the age of 16 MacMahon was admitted to the Royal Military Academy, Woolwich and passed out after two years. Military career On 12 March 1873, MacMahon was posted to Madras, India, with the 1st Battery 5th Brigade, with the temporary rank of lieutenant. The Army List showed that in October 1873 he was posted to the 8th Brigade in Lucknow. MacMahon's final posting was to the No. 1 Mountain Battery with the Punjab Frontier Force at Kohat on the North West ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Andrews (mathematician)
George Eyre Andrews (born December 4, 1938) is an American mathematician working in special functions, number theory, mathematical analysis, analysis and combinatorics. Education and career He is currently an Evan Pugh Professor of Mathematics at Pennsylvania State University. He did his undergraduate studies at Oregon State University and received his PhD in 1964 at the University of Pennsylvania where his advisor was Hans Rademacher. During 2008–2009 he was president of the American Mathematical Society. Contributions Andrews's contributions include several monographs and over 250 research and popular articles on q-series, special functions, combinatorics and applications. He is considered to be the world's leading expert in the theory of integer partitions. In 1976 he discovered Ramanujan's Ramanujan's lost notebook, Lost Notebook. He is highly interested in mathematical pedagogy. His book ''The Theory of Partitions'' is the standard reference on the subject of integer par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman Macleod Ferrers
Norman Macleod Ferrers D.D. (11 August 1829 – 31 January 1903) was a British mathematician and university administrator and editor of a mathematical journal. Career and research Ferrers was educated at Eton College before studying at Gonville and Caius College, Cambridge, where he was Senior Wrangler in 1851. He was appointed to a Fellowship at the college in 1852, was called to the bar in 1855 and was ordained deacon in 1859 and priest in 1860. In 1880, he was appointed Master of the college, and served as vice-chancellor of Cambridge University from 1884 to 1885. Ferrers made many contributions to mathematical literature. From 1855 to 1891 he worked with J. J. Sylvester as editors, with others, in publishing The Quarterly Journal of Pure and Applied Mathematics. Ferrers assembled the papers of George Green for publication in 1871. In 1861 he published "An Elementary Treatise on Trilinear Co-ordinates". One of his early contributions was on Sylvester's development of Poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." Euler is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Footnotes
A note is a string of text placed at the bottom of a page in a book or document or at the end of a chapter, volume, or the whole text. The note can provide an author's comments on the main text or citations of a reference work in support of the text. Footnotes are notes at the foot of the page while endnotes are collected under a separate heading at the end of a chapter, volume, or entire work. Unlike footnotes, endnotes have the advantage of not affecting the layout of the main text, but may cause inconvenience to readers who have to move back and forth between the main text and the endnotes. In some editions of the Bible, notes are placed in a narrow column in the middle of each page between two columns of biblical text. Numbering and symbols In English, a footnote or endnote is normally flagged by a superscripted number immediately following that portion of the text the note references, each such footnote being numbered sequentially. Occasionally, a number between brack ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enumerative Combinatorics
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets ''S''''i'' indexed by the natural numbers, enumerative combinatorics seeks to describe a ''counting function'' which counts the number of objects in ''S''''n'' for each ''n''. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. The twelvefold way provides a unified framework for counting permutations, combinations and partitions. The simplest such functions are ''closed formulas'', which can be expressed as a composition of elementary functions such as factorials, powers, and so on. For instance, as shown below, the number of different possible orderings of a deck of '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
