en>Intgr: /* top */ fix caps

2014-11-20T22:50:15Z

top: fix caps

en>Tdslk: copy edits

2014-01-10T21:40:02Z

copy edits

← Older revision		Revision as of 23:40, 10 January 2014
Line 1:		Line 1:
	~~28 yr old Software Engineer Mary~~ from ~~Saint-Hubert~~, ~~likes~~ to ~~spend some~~ time ~~creating~~, ~~Anti Depressant Drugs~~ and ~~traveling~~. ~~Has~~ in ~~recent times completed~~ a ~~journey~~ to ~~Archaeological Areas~~ of ~~Pompei~~.<br><~~br>Stop~~ by ~~my webpage~~ - [https://~~www~~.~~youtube~~.~~com~~/~~watch?v~~=~~sz__0LuxVM8 zoloft withdrawal lawsuit~~]		{{contradict-other-multiple\|Permutation\|Lehmer code\|Factorial number system\|date=March 2013}}
			[[File:Inversion set and vector of a permutation.svg\|thumb\|right\|380px\|The permutation (4,1,5,2,6,3) has the inversion vector (0,1,0,2,0,3) and the inversion set {(1,2),(1,4),(3,4),(1,6),(3,6),(5,6)}. The inversion vector [[w:Factorial number system\|converted]] to decimal is 373.]]
			[[File:Inversion set 16; wp(13,11, 7,15).svg\|thumb\|250px\|Inversion set of the permutation<br>(0,15, 14,1, 13,2, 3,12,<br>11,4, 5,10, 6,9, 8,7)<br>showing the pattern of the<br>[[Thue–Morse sequence]]]]
			In [[computer science]] and [[discrete mathematics]], an '''inversion''' is a pair of places of a sequence where the elements on these places are out of their natural [[total order\|order]].

			== Definitions ==

			Formally, let <math>(A(1), \ldots, A(n))</math> be a sequence of ''n'' distinct numbers. If <math>i < j</math> and <math>A(i) > A(j)</math>, then the pair <math>(i, j)</math> is called an inversion of <math>A</math>.{{sfn\|Cormen\|Leiserson\|Rivest\|Stein\|2001\|pp=39}}{{sfn\|Vitter\|Flajolet\|1990\|pp=459}}

			The '''inversion number''' of a sequence is one common measure of its sortedness.{{sfn\|Barth\|Mutzel\|2004\|pp=183}}{{sfn\|Vitter\|Flajolet\|1990\|pp=459}} Formally, the inversion number is defined to be the number of inversions, that is,
			:<math>\text{inv}(A) = \# \{(A(i),A(j)) \mid i < j \text{ and } A(i) > A(j)\}</math>.{{sfn\|Barth\|Mutzel\|2004\|pp=183}}
			Other measures of (pre-)sortedness include the minimum number of elements that can be deleted from the sequence to yield a fully sorted sequence, the number and lengths of sorted "runs" within the sequence, and the smallest number of exchanges needed to sort the sequence.{{sfn\|Mahmoud\|2000\|pp=284}} Standard [[comparison sort]]ing algorithms can be adapted to compute the inversion number in time {{math\|O(''n'' log ''n'')}}.

			The '''inversion vector''' ''V(i)'' of the sequence is defined for ''i'' = 2, ..., ''n'' as <math>V[i] = \left\vert\{k \mid k < i \text{ and } A(k) > A(i)\}\right\vert</math>. In other words each element is the number of elements preceding the element in the original sequence that are greater than it. Note that the inversion vector of a sequence has one less element than the sequence, because of course the number of preceding elements that are greater than the first is always zero. Each permutation of a sequence has a unique inversion vector and it is possible to construct any given permutation of a (fully sorted) sequence from that sequence and the permutation's inversion vector.{{sfn\|Pemmaraju\|Skiena\|2003\|pp=69}}

			==Weak order of permutations==
			The set of permutations on ''n'' items can be given the structure of a [[partial order]], called the '''weak order of permutations''', which forms a [[lattice (order)\|lattice]].

			To define this order, consider the items being permuted to be the integers from 1 to ''n'', and let Inv(''u'') denote the set of inversions of a permutation ''u'' for the natural ordering on these items. That is, Inv(''u'') is the set of ordered pairs (''i'', ''j'') such that 1 ≤ ''i'' < ''j'' ≤ ''n'' and ''u''(''i'') > ''u''(''j''). Then, in the weak order, we define ''u'' ≤ ''v'' whenever Inv(''u'') ⊆ Inv(''v'').

			The edges of the [[Hasse diagram]] of the weak order are given by permutations ''u'' and ''v'' such that ''u < v'' and such that ''v'' is obtained from ''u'' by interchanging two consecutive values of ''u''. These edges form a [[Cayley graph]] for the [[symmetric group\|group of permutations]] that is isomorphic to the [[skeleton (topology)\|skeleton]] of a [[permutohedron]].

			The identity permutation is the minimum element of the weak order, and the permutation formed by reversing the identity is the maximum element.

			== See also ==
			{{wikiversity\|Inversion (discrete mathematics)}}
			{{commons\|Category:Inversion (discrete mathematics)\|Inversion (discrete mathematics)}}
			* [[Factorial number system]] (a factorial number is a reflected inversion vector)
			* [[Permutation group#Transpositions, simple transpositions, inversions and sorting\|Transpositions, simple transpositions, inversions and sorting]]
			* [[Damerau–Levenshtein distance]]
			* [[Parity of a permutation]]

			'''Sequences in the [[On-Line Encyclopedia of Integer Sequences\|OEIS]]:'''
			* [https://oeis.org/wiki/Index_to_OEIS:_Section_Fa#factorial Index entries for sequences related to factorial numbers]
			* Reflected inversion vectors: {{OEIS link\|A007623}} and {{OEIS link\|A108731}}
			* Sum of inversion vectors, cardinality of inversion sets: {{OEIS link\|A034968}}
			* Inversion sets of finite permutations interpreted as binary numbers: {{OEIS link\|A211362}}   (related permutation: {{OEIS link\|A211363}})
			* Finite permutations that have only 0s and 1s in their inversion vectors: {{OEIS link\|A059590}}   (their inversion sets: {{OEIS link\|A211364}})
			* Numbers of permutations of n elements with k inversions; Mahonian numbers: {{OEIS link\|A008302}}   (their row maxima; Kendall-Mann numbers: {{OEIS link\|A000140}})
			* Number of connected labeled graphs with n edges and n nodes: {{OEIS link\|A057500}}
			* Arrays of permutations with similar inversion sets and inversion vectors: {{OEIS link\|A211365}}, {{OEIS link\|A211366}}, {{OEIS link\|A211367}}, {{OEIS link\|A211368}}, {{OEIS link\|A211369}}, {{OEIS link\|A100630}}, {{OEIS link\|A211370}}, {{OEIS link\|A051683}}

			== References ==
			{{reflist\|4\|refs=}}

			=== Source bibliography ===
			{{refbegin\|1}}
			* {{cite journal\|ref=harv\|first1=Wilhelm\|last1=Barth\|first2=Petra\|last2=Mutzel\|title=Simple and Efficient Bilayer Cross Counting\|journal=[[Journal of Graph Algorithms and Applications]]\|volume=8\|issue=2\|pages=179–194\|year=2004}}
			* {{cite book\|ref=harv
			\| first1=Thomas H.\|last1=Cormen\|authorlink1=Thomas H. Cormen
			\| last2=Leiserson\|first2=Charles E.\|authorlink2=Charles E. Leiserson
			\| last3=Rivest\|first3=Ronald L.\|authorlink3=Ron Rivest
			\| last4=Stein\|first4=Clifford\|authorlink4=Clifford Stein
			\| title = [[Introduction to Algorithms]]
			\| publisher = MIT Press and McGraw-Hill
			\| year = 2001
			\| isbn = 0-262-53196-8
			\| edition = 2nd
			}}
			* {{cite book\|ref=harv\|title=Sorting: a distribution theory\|chapter=Sorting Nonrandom Data\|volume=54\|series=Wiley-Interscience series in discrete mathematics and optimization\|first=Hosam Mahmoud\|last=Mahmoud\|publisher=Wiley-IEEE\|year=2000\|isbn=978-0-471-32710-3}}
			* {{cite book\|ref=harv\|title=Computational discrete mathematics: combinatorics and graph theory with Mathematica\|chapter=Permutations and combinations\|first1=Sriram V.\|last1=Pemmaraju\|first2=Steven S.\|last2=Skiena\|publisher=Cambridge University Press\|year=2003\|isbn=978-0-521-80686-2}}
			* {{cite book\|ref=harv\|title=Algorithms and Complexity\|volume=1\|editor1-first=Jan\|editor1-last=van Leeuwen\|editor1-link=Jan van Leeuwen\|edition=2nd\|publisher=Elsevier\|year=1990\|isbn=978-0-444-88071-0\|chapter=Average-Case Analysis of Algorithms and Data Structures\|first1=J.S.\|last1=Vitter\|first2=Ph.\|last2=Flajolet}}
			{{refend}}

			=== Further reading ===
			* {{cite journal\|ref=harv\|journal=Journal of Integer Sequences\|volume=4\|year=2001\|title=Permutations with Inversions\|first=Barbara H.\|last=Margolius}}

			=== Presortedness measures ===
			* {{cite journal\|ref=harv\|journal=Lecture Notes in Computer Science\|year=1984\|volume=172\|pages=324–336\|doi=10.1007/3-540-13345-3_29\|title=Measures of presortedness and optimal sorting algorithms\|first=Heikki\|last=Mannila\|authorlink=Heikki Mannila}}
			* {{cite journal\|ref=harv\|first1=Vladimir\|last1=Estivill-Castro\|first2=Derick\|last2=Wood\|title=A new measure of presortedness\|journal=Information and Computation\|volume=83\|issue=1\|pages=111–119\|year=1989}}
			* {{cite journal\|ref=harv\|first=Steven S.\|last=Skiena\|year=1988\|title=Encroaching lists as a measure of presortedness\|journal=BIT\|volume=28\|issue=4\|pages=755–784}}

			[[Category:Permutations]]
			[[Category:Order theory]]
			[[Category:String similarity measures]]
			[[Category:Sorting algorithms]]
			[[Category:Combinatorics]]
			[[Category:Discrete mathematics]]

75.32.232.185: /* Distribution */ Before: "The first moment is the mean distance between randomly distributed particles in $D$ dimensions"

2012-03-26T05:15:10Z

Distribution: Before: "The first moment is the mean distance between randomly distributed particles in <math>D</math> dimensions"

New page

28 yr old Software Engineer Mary from Saint-Hubert, likes to spend some time creating, Anti Depressant Drugs and traveling. Has in recent times completed a journey to Archaeological Areas of Pompei.<br><br>Stop by my webpage - [https://www.youtube.com/watch?v=sz__0LuxVM8 zoloft withdrawal lawsuit]

← Older revision		Revision as of 23:40, 10 January 2014
Line 1:		Line 1:
	~~28 yr old Software Engineer Mary~~ from ~~Saint-Hubert~~, ~~likes~~ to ~~spend some~~ time ~~creating~~, ~~Anti Depressant Drugs~~ and ~~traveling~~. ~~Has~~ in ~~recent times completed~~ a ~~journey~~ to ~~Archaeological Areas~~ of ~~Pompei~~.<br><~~br>Stop~~ by ~~my webpage~~ - [https://~~www~~.~~youtube~~.~~com~~/~~watch?v~~=~~sz__0LuxVM8 zoloft withdrawal lawsuit~~]		{{contradict-other-multiple\|Permutation\|Lehmer code\|Factorial number system\|date=March 2013}}
			[[File:Inversion set and vector of a permutation.svg\|thumb\|right\|380px\|The permutation (4,1,5,2,6,3) has the inversion vector (0,1,0,2,0,3) and the inversion set {(1,2),(1,4),(3,4),(1,6),(3,6),(5,6)}. The inversion vector [[w:Factorial number system\|converted]] to decimal is 373.]]
			[[File:Inversion set 16; wp(13,11, 7,15).svg\|thumb\|250px\|Inversion set of the permutation<br>(0,15, 14,1, 13,2, 3,12,<br>11,4, 5,10, 6,9, 8,7)<br>showing the pattern of the<br>[[Thue–Morse sequence]]]]
			In [[computer science]] and [[discrete mathematics]], an '''inversion''' is a pair of places of a sequence where the elements on these places are out of their natural [[total order\|order]].

			== Definitions ==

			Formally, let <math>(A(1), \ldots, A(n))</math> be a sequence of ''n'' distinct numbers. If <math>i < j</math> and <math>A(i) > A(j)</math>, then the pair <math>(i, j)</math> is called an inversion of <math>A</math>.{{sfn\|Cormen\|Leiserson\|Rivest\|Stein\|2001\|pp=39}}{{sfn\|Vitter\|Flajolet\|1990\|pp=459}}

			The '''inversion number''' of a sequence is one common measure of its sortedness.{{sfn\|Barth\|Mutzel\|2004\|pp=183}}{{sfn\|Vitter\|Flajolet\|1990\|pp=459}} Formally, the inversion number is defined to be the number of inversions, that is,
			:<math>\text{inv}(A) = \# \{(A(i),A(j)) \mid i < j \text{ and } A(i) > A(j)\}</math>.{{sfn\|Barth\|Mutzel\|2004\|pp=183}}
			Other measures of (pre-)sortedness include the minimum number of elements that can be deleted from the sequence to yield a fully sorted sequence, the number and lengths of sorted "runs" within the sequence, and the smallest number of exchanges needed to sort the sequence.{{sfn\|Mahmoud\|2000\|pp=284}} Standard [[comparison sort]]ing algorithms can be adapted to compute the inversion number in time {{math\|O(''n'' log ''n'')}}.

			The '''inversion vector''' ''V(i)'' of the sequence is defined for ''i'' = 2, ..., ''n'' as <math>V[i] = \left\vert\{k \mid k < i \text{ and } A(k) > A(i)\}\right\vert</math>. In other words each element is the number of elements preceding the element in the original sequence that are greater than it. Note that the inversion vector of a sequence has one less element than the sequence, because of course the number of preceding elements that are greater than the first is always zero. Each permutation of a sequence has a unique inversion vector and it is possible to construct any given permutation of a (fully sorted) sequence from that sequence and the permutation's inversion vector.{{sfn\|Pemmaraju\|Skiena\|2003\|pp=69}}

			==Weak order of permutations==
			The set of permutations on ''n'' items can be given the structure of a [[partial order]], called the '''weak order of permutations''', which forms a [[lattice (order)\|lattice]].

			To define this order, consider the items being permuted to be the integers from 1 to ''n'', and let Inv(''u'') denote the set of inversions of a permutation ''u'' for the natural ordering on these items. That is, Inv(''u'') is the set of ordered pairs (''i'', ''j'') such that 1 ≤ ''i'' < ''j'' ≤ ''n'' and ''u''(''i'') > ''u''(''j''). Then, in the weak order, we define ''u'' ≤ ''v'' whenever Inv(''u'') ⊆ Inv(''v'').

			The edges of the [[Hasse diagram]] of the weak order are given by permutations ''u'' and ''v'' such that ''u < v'' and such that ''v'' is obtained from ''u'' by interchanging two consecutive values of ''u''. These edges form a [[Cayley graph]] for the [[symmetric group\|group of permutations]] that is isomorphic to the [[skeleton (topology)\|skeleton]] of a [[permutohedron]].

			The identity permutation is the minimum element of the weak order, and the permutation formed by reversing the identity is the maximum element.

			== See also ==
			{{wikiversity\|Inversion (discrete mathematics)}}
			{{commons\|Category:Inversion (discrete mathematics)\|Inversion (discrete mathematics)}}
			* [[Factorial number system]] (a factorial number is a reflected inversion vector)
			* [[Permutation group#Transpositions, simple transpositions, inversions and sorting\|Transpositions, simple transpositions, inversions and sorting]]
			* [[Damerau–Levenshtein distance]]
			* [[Parity of a permutation]]

			'''Sequences in the [[On-Line Encyclopedia of Integer Sequences\|OEIS]]:'''
			* [https://oeis.org/wiki/Index_to_OEIS:_Section_Fa#factorial Index entries for sequences related to factorial numbers]
			* Reflected inversion vectors: {{OEIS link\|A007623}} and {{OEIS link\|A108731}}
			* Sum of inversion vectors, cardinality of inversion sets: {{OEIS link\|A034968}}
			* Inversion sets of finite permutations interpreted as binary numbers: {{OEIS link\|A211362}}   (related permutation: {{OEIS link\|A211363}})
			* Finite permutations that have only 0s and 1s in their inversion vectors: {{OEIS link\|A059590}}   (their inversion sets: {{OEIS link\|A211364}})
			* Numbers of permutations of n elements with k inversions; Mahonian numbers: {{OEIS link\|A008302}}   (their row maxima; Kendall-Mann numbers: {{OEIS link\|A000140}})
			* Number of connected labeled graphs with n edges and n nodes: {{OEIS link\|A057500}}
			* Arrays of permutations with similar inversion sets and inversion vectors: {{OEIS link\|A211365}}, {{OEIS link\|A211366}}, {{OEIS link\|A211367}}, {{OEIS link\|A211368}}, {{OEIS link\|A211369}}, {{OEIS link\|A100630}}, {{OEIS link\|A211370}}, {{OEIS link\|A051683}}

			== References ==
			{{reflist\|4\|refs=}}

			=== Source bibliography ===
			{{refbegin\|1}}
			* {{cite journal\|ref=harv\|first1=Wilhelm\|last1=Barth\|first2=Petra\|last2=Mutzel\|title=Simple and Efficient Bilayer Cross Counting\|journal=[[Journal of Graph Algorithms and Applications]]\|volume=8\|issue=2\|pages=179–194\|year=2004}}
			* {{cite book\|ref=harv
			\| first1=Thomas H.\|last1=Cormen\|authorlink1=Thomas H. Cormen
			\| last2=Leiserson\|first2=Charles E.\|authorlink2=Charles E. Leiserson
			\| last3=Rivest\|first3=Ronald L.\|authorlink3=Ron Rivest
			\| last4=Stein\|first4=Clifford\|authorlink4=Clifford Stein
			\| title = [[Introduction to Algorithms]]
			\| publisher = MIT Press and McGraw-Hill
			\| year = 2001
			\| isbn = 0-262-53196-8
			\| edition = 2nd
			}}
			* {{cite book\|ref=harv\|title=Sorting: a distribution theory\|chapter=Sorting Nonrandom Data\|volume=54\|series=Wiley-Interscience series in discrete mathematics and optimization\|first=Hosam Mahmoud\|last=Mahmoud\|publisher=Wiley-IEEE\|year=2000\|isbn=978-0-471-32710-3}}
			* {{cite book\|ref=harv\|title=Computational discrete mathematics: combinatorics and graph theory with Mathematica\|chapter=Permutations and combinations\|first1=Sriram V.\|last1=Pemmaraju\|first2=Steven S.\|last2=Skiena\|publisher=Cambridge University Press\|year=2003\|isbn=978-0-521-80686-2}}
			* {{cite book\|ref=harv\|title=Algorithms and Complexity\|volume=1\|editor1-first=Jan\|editor1-last=van Leeuwen\|editor1-link=Jan van Leeuwen\|edition=2nd\|publisher=Elsevier\|year=1990\|isbn=978-0-444-88071-0\|chapter=Average-Case Analysis of Algorithms and Data Structures\|first1=J.S.\|last1=Vitter\|first2=Ph.\|last2=Flajolet}}
			{{refend}}

			=== Further reading ===
			* {{cite journal\|ref=harv\|journal=Journal of Integer Sequences\|volume=4\|year=2001\|title=Permutations with Inversions\|first=Barbara H.\|last=Margolius}}

			=== Presortedness measures ===
			* {{cite journal\|ref=harv\|journal=Lecture Notes in Computer Science\|year=1984\|volume=172\|pages=324–336\|doi=10.1007/3-540-13345-3_29\|title=Measures of presortedness and optimal sorting algorithms\|first=Heikki\|last=Mannila\|authorlink=Heikki Mannila}}
			* {{cite journal\|ref=harv\|first1=Vladimir\|last1=Estivill-Castro\|first2=Derick\|last2=Wood\|title=A new measure of presortedness\|journal=Information and Computation\|volume=83\|issue=1\|pages=111–119\|year=1989}}
			* {{cite journal\|ref=harv\|first=Steven S.\|last=Skiena\|year=1988\|title=Encroaching lists as a measure of presortedness\|journal=BIT\|volume=28\|issue=4\|pages=755–784}}

			[[Category:Permutations]]
			[[Category:Order theory]]
			[[Category:String similarity measures]]
			[[Category:Sorting algorithms]]
			[[Category:Combinatorics]]
			[[Category:Discrete mathematics]]

Complete spatial randomness - Revision history

en>Intgr: /* top */ fix caps

en>Tdslk: copy edits

75.32.232.185: /* Distribution */ Before: "The first moment is the mean distance between randomly distributed particles in D dimensions"

75.32.232.185: /* Distribution */ Before: "The first moment is the mean distance between randomly distributed particles in $D$ dimensions"