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Partition Theory

Partition theory is a fundamental area of number theory. It is concerned with the number of ways that a whole number can be partitioned into whole number parts.

5 for example can be partitioned in 7 ways thus:
11111, 2111, 221, 311, 32, 41, 5.
The permutations of these 7 partitions add up to 16 thus:
11111 =1 permutation
2111 =4 permutations
221 =3 permutations
311 =3 permutations
32 =2 permutations
41 =2 permutations
5 =1 permutation


A partition is a way of writing an integer n as a sum of positive integers where the order of the parts is not significant.
Number Partitions Order counts
0 1 1
1 1 1
2 2 2
3 3 4
4 5 8
5 7 16
6 11 32
7 15 64
8 22 128
9 30 256
10 42 512
11 56 1024
12 77 2048
13 101 4096
14 135 8192
15 176 16384
16 231 -
17 297 -
18 385 -
19 490 -
20 627 -
21 792 -
22 1002 -
http://home.att.net/~numericana/data/partition.htm Partition Numbers for 0 to 4096 (site down?)

5, 7, 11 Congruences: Properties and Meanings

Starting with 4, the number of partitions for every 5th integer is a multiple of 5-eg the number of partitions for 9 is 30 and for 14 is 135.

Starting with 5, every 7th integer is a multiple of 7.

Starting with 6, every 11th integer is a multiple of 11.

Other Relationships: Properties and Meanings

Number of partitions of n into an even number of distinct parts equals the number of partitions of n into an odd number of distinct parts-except when n is a Pentagonal Number.(Franklin?)

Number of partitions of n into parts not divisible by 3>than the number of partitions of n into parts none of which occurs more than twice.

The number of odd partitions of n=the number of distinct partitions of n.

The total number of 1s that occur among all unordered partitions of a positive integer is equal to the sum of the numbers of distinct members of those partitions.

SEQUENCE: 0, 1, 3, 6, 12, 20, 35, 54, 86, 128, 192, 275, 399, 556, 780, 1068, 1463, 1965, 2644, 3498, 4630, 6052, 7899, 10206, 13174, 16851, 21522, 27294, 34545, 43453, 54563, 68135, 84927, 105366, 130462, 160876, 198014, 242812, 297201, 362587, 441546, 536104, 649791, 785437, 947812, 1140945, 1371173, 1644136, 1968379, 2351597, 2805218, 3339869, 3970648, 4712040, 5584141, 6606438, 7805507, 9207637 ...
The total number of parts in all partitions of n. Also sum of all large parts of all partitions of n.

SEQUENCE: 0, 0, 1, 3, 7, 13, 24, 39, 64, 98, 150, 219, 322, 455, 645, 892, 1232, 1668, 2259, 3008, 4003, 5260, 6897, 8951, 11599, 14893, 19086, 24284, 30827, 38888, 48959, 61293, 76578, 95223, 118152, 145993, 180037, 221175, 271186, 331402, 404208, 491521...
The number of "+" signs needed to write the partitions of n.

Number of partitions of n into distinct parts >= 2....number of partitions of n into odd parts.
1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 142, 165, 192, 222, 256, 296, 340, 390, 448, 512, 585, 668, 760, 864, 982, 1113, 1260, 1426, 1610, 1816, 2048, 2304, 2590, 2910, 3264, 3658, 4097, 4582, 5120, 5718
http://www.research.att.com/~njas/sequences/?q=distinct+parts&sort=0&fmt=0&language=english&go=Search



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Results for partition theory

The Theory of Partitions (Encyclopedia of Mathematics and its Applications)
by George E. AndrewsSearch for George E. Andrews
$64.99
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This book develops the theory of partitions. Simply put, the partitions of a number are the ways of writing that number as sums of positive integers. For example, the five partitions of 4 are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. Surprisingly, such a simple matter requires some deep mathematics for its study. This book considers the many theoretical aspects of this subject, which have in turn recently found applications to statistical mechanics, computer science and other branches of mathematics. With minimal prerequisites, this book is suitable for students as well as researchers in combinatorics, analysis, and number theory.
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On Partition Theory
by Sabuj DasSearch for Sabuj Das
$89.00
$83.70 new/used

A partition of n is a division of n into any number of positive integral parts. Then the sum of the integral parts is n. The order of the parts and arrangement in a division of n are irrelevant and the parts are arranged in descending order. We denote the number of partitions of n by P(n). We introduce a brief survey of Ramanujan?s partitions congruences. We mainly concern with combinatorial aspects of the congruences moduli 5, 7 in terms of the rank and crank of vector partitions separately.
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An Introduction to the Theory of Numbers
by G. H. Hardy, Edward M. Wright, Andrew Wiles
$66.00
$28.32 new/used

An Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. Developed under the guidance of D. R. Heath-Brown, this Sixth Edition of An Introduction to the Theory of Numbers has been extensively revised and updated to guide today's students through the key milestones and developments in number theory.

Updates include a chapter by J. H. Silverman on one of the most important developments in number theory - modular elliptic curves and their role in the proof of Fermat's Last Theorem -- a foreword by A. Wiles, and comprehensively updated end-of-chapter notes detailing the key developments in number theory. Suggestions for further reading are also included for the more avid reader.

The text retains the style and clarity of previous editions making it highly suitable for undergraduates in mathematics from the first year upwards as well as an essential reference for all number theorists.
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Elementary Number Theory with Programming
by Marty Lewinter, Jeanine Meyer
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A successful presentation of the fundamental concepts of number theory and computer programming

Bridging an existing gap between mathematics and programming, Elementary Number Theory with Programming provides a unique introduction to elementary number theory with fundamental coverage of computer programming. Written by highly-qualified experts in the fields of computer science and mathematics, the book features accessible coverage for readers with various levels of experience and explores number theory in the context of programming without relying on advanced prerequisite knowledge and concepts in either area.

Elementary Number Theory with Programming features comprehensive coverage of the methodology and applications of the most well-known theorems, problems, and concepts in number theory. Using standard mathematical applications within the programming field, the book presents triangle numbers and prime decomposition, which are the basis of the public-private key system of cryptography. In addition, the book includes:

  • Numerous examples,  exercises, and research challenges in each chapter to encourage readers to work through the discussed concepts and ideas 
  • Select solutions to the chapter exercises in an appendix
  • Plentiful sample computer programs to aid comprehension of the presented material for readers who have either never done any programming or need to improve their existing skill set
  •  A related website with links to select exercises
  • An Instructor?s Solutions Manual available on a companion website
Elementary Number Theory with Programming is a useful textbook for undergraduate and graduate-level students majoring in mathematics or computer science, as well as an excellent supplement for teachers and students who would like to better understand and appreciate number theory and computer programming. The book is also an ideal reference for computer scientists, programmers, and researchers interested in the mathematical applications of programming.

 


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Schaum's Outline of Theory and Problems of Combinatorics including concepts of Graph Theory
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Exodus
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Exodus is an international publishing phenomenon--the towering novel of the twentieth century's most dramatic geopolitical event.  Leon Uris magnificently portrays the birth of a new nation in the midst of enemies--the beginning of an earthshaking struggle for power.  Here is the tale that swept the world with its fury: the story of an American nurse, an Israeli freedom fighter caught up in a glorious, heartbreaking, triumphant era.  Here is Exodus --one of the great best-selling novels of all time.
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Remembering Partition: Violence, Nationalism and History in India (Contemporary South Asia)
by Gyanendra PandeySearch for Gyanendra Pandey
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Gyan Pandey's latest book is a compelling examination of the violence that marked the partition of India in 1947, and how the preceding events have been documented. In the process, the author provides a critique of history-writing and nationalist myth-making. He also investigates how local forms of community are established by the way in which violent events are remembered and written about. The book will be of interest to historians of South Asia, to sociologists and to anyone concerned with the Indian subaltern story.
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Combinatorial Set Theory: Partition Relations for Cardinals
by Paul ErdosSearch for Paul Erdos
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This work presents the most important combinatorial ideas in partition calculus and discusses ordinary partition relations for cardinals without the assumption of the generalized continuum hypothesis. A separate section of the book describes the main partition symbols scattered in the literature. A chapter on the applications of the combinatorial methods in partition calculus includes a section on topology with Arhangel'skii's famous result that a first countable compact Hausdorff space has cardinality, at most continuum. Several sections on set mappings are included as well as an account of recent inequalities for cardinal powers that were obtained in the wake of Silver's breakthrough result saying that the continuum hypothesis can not first fail at a singular cardinal of uncountable cofinality.
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The Longest August: The Unflinching Rivalry Between India and Pakistan
by Dilip HiroSearch for Dilip Hiro
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The partitioning of British India into independent Pakistan and India in August 1947 occurred in the midst of communal holocaust, with Hindus and Sikhs on one side and Muslims on the other. More than 750,000 people were butchered, and 12 million fled their homes—primarily in caravans of bullock-carts—to seek refuge across the new border: it was the largest exodus in history. Sixty-seven years later, it is as if that August never ended.

Renowned historian and journalist Dilip Hiro provides a riveting account of the relationship between India and Pakistan, tracing the landmark events that led to the division of the sub-continent and the evolution of the contentious relationship between Hindus and Muslims. To this day, a reasonable resolution to their dispute has proved elusive, and the Line of Control in Kashmir remains the most heavily fortified frontier in the world, with 400,000 soldiers arrayed on either side.

Since partition, there have been several acute crises between the neighbors, including the secession of East Pakistan to form an independent Bangladesh in 1971, and the acquisition of nuclear weapons by both sides resulting in a scarcely avoided confrontation in 1999 and again in 2002. Hiro amply demonstrates the geopolitical importance of the India-Pakistan conflict by chronicling their respective ties not only with America and the Soviet Union, but also with China, Israel, and Afghanistan.

Hiro weaves these threads into a lucid narrative, enlivened with colorful biographies of leaders, vivid descriptions of wars, sensational assassinations, gross violations of human rights—and cultural signifiers like cricket matches. The Longest August is incomparable in its scope and presents the first definitive history of one of the world?s longest-running and most intractable conflicts.

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