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Topic: Partition of an integer

	PlanetMath: integer partition
		The dual partition is the partition obtained by reflecting the Young diagram along the main diagonal.
		This is version 4 of integer partition, born on 2004-04-10, modified 2006-06-09.
		(Combinatorics :: Enumerative combinatorics :: Partitions of integers)
	planetmath.org /encyclopedia/IntegerPartition.html (169 words)

	Info on Numerical Partitions
		The number of partitions of n is denoted p(n) and the number of partitions of n into k parts is denoted p(n,k).
		A Ferrers diagram is a pictorial representation of a partition.
		A000009 in Neil J. Sloane's database of integer sequences.
	www.theory.csc.uvic.ca /~cos/inf/nump/NumPartition.html (594 words)

	Integer partition: Definition and Links by Encyclopedian.com
		In mathematics, a partition of a positive integer n is a way of writing n as a sum of positive integers.
		In the case of the number 4, partitions 4 and 1 + 1 + 1 + 1 are conjugate pairs, and partitions 3 + 1 and 2 + 1 + 1 are conjugate of each other.
		The number of partitions of a positive integer n is given by the Partition function p(n).
	www.encyclopedian.com /in/Integer-partition.html (793 words)

	Generating Partitions
		For example, the seven distinct integer partitions of 5 are {5}, {4,1}, {3,2}, {3,1,1}, {2,2,1}, {2,1,1,1}, and {1,1,1,1,1}.
		An interesting application I encountered that required the generation of integer partitions was in a simulation of nuclear fission.
		Young tableaux are two-dimensional configurations of integers {1,...,n} where the number of elements in each row is defined by an integer partition of n.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE153.HTM (1138 words)

	Integer partitioning problems - connections with physics
		In either case one is concerned with partitioning a large integer, under certain restrictions, which in effect means that the 'Zustandsumme' of a thermodynamic assembly is identical with the generating function of partitions appropriate to that assembly.
		An interesting example is provided in a recent paper of Agarwala and Auluck where the Hardy-Ramanujan formula for partitions into integral powers of integers is shown to be valid for non-integral powers as well.
		This classic integer programming problem consists of partitioning a sequence of N positive real numbers $\{a_1, a_2,..., a_N}$ (the instance) into two sets such that the absolute value of the difference of the sums of $a_j$ over the two sets is minimized.
	www.secamlocal.ex.ac.uk /~mwatkins/zeta/partitioning.htm (2994 words)

	Information on Binary Partitions (Site not responding. Last check: 2007-11-05)
		A binary partition of an integer m is a is a numerical partition of m, all of whose parts are powers of 2.
		For example, the four binary partitions of m = 5 are 1 1 1 1 1, 1 1 1 2, 1 2 2, and 1 4.
		A000123 in Neil J. Sloane's database of integer sequences.
	www.theory.cs.uvic.ca /~cos/inf/nump/BinaryPartition.html (186 words)

	NumPart
		A numerical partition of an integer n is a sequence of integers called parts whose sum is n.
		The number of partitions of a given number n is denoted by p(n).
		This is formed by an arrangement of n dots on a grid where each part in the partition is represented by placing the same number of dots in a row.
	home.snu.edu /~lturner/MC-MathStr/NumPart.htm (258 words)

	[No title]
		To change the partition's group, the calling process must either be the system administrator or must be the partition's owner and changing the group to a group that the calling process belongs to.
		A partition's protection mode consists of three groups of permission bits that indicate the read, write and execute permissions for the owner, group, and other users of the partition.
		Partition lock denied The specified partition is currently being updated and is locked by someone else.
	www.sandia.gov /ASCI/Red/usage/paragon/man/man3/nx_chpart.3f.html (896 words)

	Search ScienceWorld
		The number of graphical partitions on n-node graphs is therefore the same as the number of n-node graphs with no isolated points, i.e., 0, 1, 2, 7, 23, 122, 888, 11302, 262322,...
		A partition is a way of writing an integer n as a sum of positive integers where the order of the addends is not significant, possibly subject to one or more additional constraints.
		The Ferrers diagrams corresponding to the self-conjugate partitions for 3
	scienceworld.wolfram.com /search/index.cgi?as_q=Partition+of+Numbers (478 words)

	[No title]
		Integer Partitions ------------------ By an _integer partition_ of a natural number n we mean a nonincreasing finite sequence (a_1,...
		Given an integer partitions, we call the numbers a_i _parts_ of the partition.
		EXERCISE: Justify this by an induction modeled on the preceding argument.
	www.math.unh.edu /~dvf/532/Integer_Partitions (541 words)

	Partition (number theory) - Wikipedia, the free encyclopedia
		In number theory, a partition of a positive integer n is a way of writing n as a sum of positive integers.
		In number theory, the partition function p(n) represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers.
		For instance, the number of partitions for the integer 4 is 5.
	en.wikipedia.org /wiki/Integer_partition (1501 words)

	The On-Line Encyclopedia of Integer Sequences
		Number of one-element transitions among partitions of the integer n for unlabeled parts.
		a(n) = Sum_i=1^P(n+1) S(i, n+1)^2 - S(i, n+1), where P(n+1) is the number of integer partitions of n+1 and S(i, n+1) is the number of digits in the set of digits of the i-th partition of n+1.
		In the unlabeled case we have 10 one-element transitions among all partitions of n=4: [1,1,1,1] -> [1,1,2]; [1,1,2] -> [2,2]; [1,1,2] -> [1,3]; [2,2] -> [1,3]; [1,3] -> [4] and [1,1,2] -> [1,1,1,1]; [2,2] -> [1,1,2]; [1,3] -> [1,1,2]; [1,3] -> [2,2]; [4] -> [1,3].
	www.research.att.com /~njas/sequences/A093695 (441 words)

	Nabble - Can anyone help me with partition numbers?
		A partition of a positive integer n is a representation of n as the sum of any number of positive integral parts.
		For example, there are 7 partitions of the number 5: 1+1+1+1+1, 1+1+1+2, 1+1+3, 1+2+2, 1+4, 2+3 and 5.
		> A partition of a positive integer n is a representation of n as the sum of
	www.nabble.com /Can-anyone-help-me-with-partition-numbers--t612336.html (331 words)

	PlanetMath: part of a partition
		"part of a partition" is owned by drini.
		This is version 2 of part of a partition, born on 2005-02-10, modified 2005-02-12.
		In this case, I would suggest changing the title of the entry to something much more concrete and related to the entry, like "part of a partition" or similar.
	planetmath.org /encyclopedia/Part.html (111 words)

	List of partition topics - Wikipedia, the free encyclopedia
		For the political sense of the word partition see for example: history of Cyprus, history of Ireland, partition of India, partitions of Poland, 1947 UN Partition Plan (Palestine).
		The partition disambiguation page lists meanings in other fields as well.
		a partition of the sum of squares in statistics problems, especially in the analysis of variance.
	en.wikipedia.org /wiki/List_of_partition_topics (129 words)

	Undergraduate Spends Summer in Research (Site not responding. Last check: 2007-11-05)
		A partition of a nonnegative integer n is a representation of n as a sum of positive integers, called summands, the order of which are irrelevant.
		The number of these partitions for an integer n is denoted by the partition function, p(n).
		(n), which is the partition of an integer into summands none of which are divisible by k.
	www.unomaha.edu /wwwmath/OurArchive/NEWS2001/summerresearch.html (514 words)

	CS146 Program Assignment 2 (Site not responding. Last check: 2007-11-05)
		For example, the partitions of four, given by (1, 1, 1, 1), (1, 1, 2), (2, 2), (4), and (1, 3) correspond to the solutions(j
		Given an integer n and an integer p which is smaller than n, we want to generate all partition into p parts of the form (1,2,a
		A partition of an integer, n, is a set of positive integers, n
	www.mathcs.sjsu.edu /faculty/lee/cs146/24SpCS146ProgramAssignment2.htm (1015 words)

	Partitions
		A partition of a positive integer n is a decreasing sequence [n_1, n_2,..., n_k] of positive integers such that sum(n_i)=n.
		Given a sequence of positive integers P which is a partition, return a positive integer which is the sum of it's parts.
		Given a sequence of positive integers P which is a partition, return its lexicographical order among partitions of the same weight.
	www.umich.edu /~gpcc/scs/magma/text1172.htm (427 words)

	DBAzine.com: Integer Labeling in Nested Intervals Model
		Given that a set of all pairs of integer numbers can be enumerated, it easily follows that we can code each node with one integer only.
		An immediate property of the previously mentioned labeling schema is density: all integer positive numbers are utilized.
		is an ancestor of node 3, which is a sibling of node 1 (if we are allowed to consider the roots of the forest as siblings).
	www.dbazine.com /oracle/or-articles/tropashko6 (1536 words)

	WORMS Brian's Digest: Enumeration
		By "partition of an integer", I mean a set of integers that sums to n.
		Generally, partitioning problems require that the number of partitions be specified and usually, the order must be considered.
		I was looking for any hints or refs to an efficient algorithm to generate the 3x2x3 possible sets each containing a unique element from A, a unique element from B and a unique element from C (i.e., an algorithm to generate {a1,b1,c1}, {a1,b1,c2},..., {a3,b2,c3}).
	www.worms.ms.unimelb.edu.au /digest/enumeration.html (908 words)

	SUBSET - Combinatorial Routines
		i4_partition_count.m, returns the number of partitions of an integer.
		i4_partition_count2.m, returns the number of partitions of an integer.
		i4vec_reverse.m, reverses the elements of an integer vector.
	www.csit.fsu.edu /~burkardt/m_src/subset/subset.html (2257 words)

	Partition -- from Wolfram MathWorld
		as a sum of positive integers where the order of the addends is not significant, possibly subject to one or more additional constraints.
		Particular types of partition functions include the partition function P, giving the number of partitions of a number as a sum of smaller integers without regard to order, and
		positive integers without regard to order and with the constraint that all integers in each sum are distinct.
	mathworld.wolfram.com /Partition.html (408 words)

	numcom22 (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-11-05)
		A partition of an integer is a representation of that integer as a sum of positive integers.
		The number of partitions of an integer n will be denoted by p(n).
		The order of the summands is irrelevant, that is, the partition 2 + 3 is not considered to be different from 3 + 2.
	www.eng.um.edu.mt.cob-web.org:8888 /~andebo/numbers/numcom22.htm (393 words)

	PlanetMath: partition function
		is defined to be the number of partitions of the integer
		Hardy and E. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 2003.
		This is version 6 of partition function, born on 2006-06-09, modified 2006-09-30.
	planetmath.org /encyclopedia/PartitionFunction2.html (139 words)

	GAP Manual: 46.14. Partitions (Site not responding. Last check: 2007-11-05)
		An unordered partition is an unordered sum n = p_1+p_2 +..+ p_k of positive integers and is represented by the list p = [p_1,p_2,..,p_k], in nonincreasing order, i.e., p_1>=p_2>=..>=p_k.
		It is possible to associate with every partition of the integer
		much larger than 40, in which case there are 37338 partitions, since the list will simply become too large.
	www.math.uiuc.edu /Software/GAP-Manual/Partitions.html (167 words)

	CLHS: System Class INTEGER
		There is no limit on the magnitude of an integer.
		The types fixnum and bignum form an exhaustive partition of type integer.
		This denotes the integers on the interval described by lower-limit and upper-limit.
	www.lisp.org /HyperSpec/Body/syscla_integer.html (86 words)

	1.3.6 Generating Partitions (Site not responding. Last check: 2007-11-05)
		Excerpt from The Algorithm Design Manual: There are two different types of combinatorial objects denoted by the term ``partition'', namely integer partitions and set partitions.
		An interesting application I encountered that required the generation of integer partitions was in a simulation of nuclear fission.
		When an atom is smashed, the nucleus of protons and neutrons is broken into a set of smaller clusters.
	www.cs.sunysb.edu /~algorith/files/generating-partitions.shtml (306 words)

	[No title]
		After creating a partition, you are the partition's owner and you can use the nx_chpart...() functions or the chpart command to change the partition's characteristics.
		This limit does not affect the priority of applications or partitions within a parti- tion.
		You cannot change a partition's scheduling to gang scheduling if the request exceeds the maximum number of partitions allocated for gang scheduling.
	www.sandia.gov /ASCI/Red/usage/paragon/man/man3/nx_chpart_epl.3f.html (1148 words)

	COMBO - Kreher and Stinson Combinatorial Routines
		Routines are available to count, list, rank and unrank such objects as balanced sequences, trees, subsets, K subsets, partitions, restricted growth functions, and permutations.
		IVEC_REVERSE reverses the elements of an integer vector.
		PART_ENUM enumerates the number of partitions of N. checks a reverse standard form partition of an integer.
	orion.math.iastate.edu /burkardt/f_src/combo/combo.html (756 words)

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