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| {{contradict-other-multiple|Permutation|Lehmer code|Factorial number system|date=March 2013}}
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| [[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.]]
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| [[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]]]]
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| 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]].
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| == Definitions ==
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| 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}}
| | Antidepressant drugs for treatment of depression as well as other disorders that may occur alone or in combination with depression, such as chronic pain, sleep disorders, or anxiety disorders. If you are taking an antidepressant, especially if you are under 25, let your doctor know if your depression seems to be getting worse or if you have any thoughts of hurting yourself. If you and your doctor decide that you should try medication for your depression, know that it takes about six weeks for the depression medication to have its full effect. Be sure to discuss any side effects you experience with your doctor.<br><br>A major 2006 U.S. government study showed that fewer than 50 percent of people become symptom-free on antidepressants, even after trying two different medications. Other studies show that the benefits of depression medication have been exaggerated, with some researchers concluding that—when it comes to mild to moderate depression—antidepressants are only slightly more effective than placebos. So while drug treatment can be beneficial, it's by no means the only answer.<br><br>Spine-Health publishes original articles written for patients by over 100 physician authors and peer-reviewed by a 17 member Medical Advisory Board This trusted, independent site is supported by hundreds of physician members and visited by millions of patients and their physicians. The information in is not intended as a [http://Search.Un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=substitute&Submit=Go substitute] for medical professional help or advice but is to be used only as an aid in understanding back pain. A depression can range in intensity from mild to severe.<br><br>While some people have one episode of depression and then never have another, or remain symptom-free for years, others have more frequent episodes or very long-lasting depressions that may go on for years. Some people find that their depressions become more frequent and severe as they get older. For these people, continuing (maintenance) treatment with antidepressants can be an effective way of reducing the frequency and [http://Search.About.com/?q=severity severity] of depressions. But, they do not work in everybody. <br><br>More than 60% of Americans taking antidepressant medication have taken it for 2 years or longer, with 14% having taken the medication for 10 years or more. Less than one-third of Americans taking one antidepressant medication and less than one-half of those taking multiple antidepressants have seen a mental health professional in the past year. This data brief discusses all antidepressants taken, regardless of the reason for use. Pharmacoepidemiol Drug Saf 16(5):560-70.<br><br>If you enjoyed this short article and you would such as to get additional info concerning [https://www.youtube.com/watch?v=sz__0LuxVM8 zoloft withdrawal lawsuit] kindly browse through the web page. |
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| 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,
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| :<math>\text{inv}(A) = \# \{(A(i),A(j)) \mid i < j \text{ and } A(i) > A(j)\}</math>.{{sfn|Barth|Mutzel|2004|pp=183}}
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| 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'')}}.
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| 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}}
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| ==Weak order of permutations==
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| 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]].
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| 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'').
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| 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]].
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| The identity permutation is the minimum element of the weak order, and the permutation formed by reversing the identity is the maximum element.
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| == See also ==
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| {{wikiversity|Inversion (discrete mathematics)}}
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| {{commons|Category:Inversion (discrete mathematics)|Inversion (discrete mathematics)}}
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| * [[Factorial number system]] (a factorial number is a reflected inversion vector)
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| * [[Permutation group#Transpositions, simple transpositions, inversions and sorting|Transpositions, simple transpositions, inversions and sorting]]
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| * [[Damerau–Levenshtein distance]]
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| * [[Parity of a permutation]]
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| '''Sequences in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]:'''
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| * [https://oeis.org/wiki/Index_to_OEIS:_Section_Fa#factorial Index entries for sequences related to factorial numbers]
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| * Reflected inversion vectors: {{OEIS link|A007623}} and {{OEIS link|A108731}}
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| * Sum of inversion vectors, cardinality of inversion sets: {{OEIS link|A034968}}
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| * Inversion sets of finite permutations interpreted as binary numbers: {{OEIS link|A211362}} (related permutation: {{OEIS link|A211363}})
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| * Finite permutations that have only 0s and 1s in their inversion vectors: {{OEIS link|A059590}} (their inversion sets: {{OEIS link|A211364}})
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| * Numbers of permutations of n elements with k inversions; Mahonian numbers: {{OEIS link|A008302}} (their row maxima; Kendall-Mann numbers: {{OEIS link|A000140}})
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| * Number of connected labeled graphs with n edges and n nodes: {{OEIS link|A057500}}
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| * 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}}
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| == References == | |
| {{reflist|4|refs=}}
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| === Source bibliography === | |
| {{refbegin|1}}
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| * {{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}}
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| * {{cite book|ref=harv
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| | first1=Thomas H.|last1=Cormen|authorlink1=Thomas H. Cormen
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| | last2=Leiserson|first2=Charles E.|authorlink2=Charles E. Leiserson
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| | last3=Rivest|first3=Ronald L.|authorlink3=Ron Rivest
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| | last4=Stein|first4=Clifford|authorlink4=Clifford Stein
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| | title = [[Introduction to Algorithms]]
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| | publisher = MIT Press and McGraw-Hill
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| | year = 2001
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| | isbn = 0-262-53196-8
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| | 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}}
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| * {{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}}
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| * {{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}}
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| {{refend}}
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| === Further reading ===
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| * {{cite journal|ref=harv|journal=Journal of Integer Sequences|volume=4|year=2001|title=Permutations with Inversions|first=Barbara H.|last=Margolius}}
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| === Presortedness measures ===
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| * {{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}}
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| * {{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}}
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| * {{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}}
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| [[Category:Permutations]]
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| [[Category:Order theory]] | |
| [[Category:String similarity measures]]
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| [[Category:Sorting algorithms]]
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| [[Category:Combinatorics]]
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| [[Category:Discrete mathematics]]
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Antidepressant drugs for treatment of depression as well as other disorders that may occur alone or in combination with depression, such as chronic pain, sleep disorders, or anxiety disorders. If you are taking an antidepressant, especially if you are under 25, let your doctor know if your depression seems to be getting worse or if you have any thoughts of hurting yourself. If you and your doctor decide that you should try medication for your depression, know that it takes about six weeks for the depression medication to have its full effect. Be sure to discuss any side effects you experience with your doctor.
A major 2006 U.S. government study showed that fewer than 50 percent of people become symptom-free on antidepressants, even after trying two different medications. Other studies show that the benefits of depression medication have been exaggerated, with some researchers concluding that—when it comes to mild to moderate depression—antidepressants are only slightly more effective than placebos. So while drug treatment can be beneficial, it's by no means the only answer.
Spine-Health publishes original articles written for patients by over 100 physician authors and peer-reviewed by a 17 member Medical Advisory Board This trusted, independent site is supported by hundreds of physician members and visited by millions of patients and their physicians. The information in is not intended as a substitute for medical professional help or advice but is to be used only as an aid in understanding back pain. A depression can range in intensity from mild to severe.
While some people have one episode of depression and then never have another, or remain symptom-free for years, others have more frequent episodes or very long-lasting depressions that may go on for years. Some people find that their depressions become more frequent and severe as they get older. For these people, continuing (maintenance) treatment with antidepressants can be an effective way of reducing the frequency and severity of depressions. But, they do not work in everybody.
More than 60% of Americans taking antidepressant medication have taken it for 2 years or longer, with 14% having taken the medication for 10 years or more. Less than one-third of Americans taking one antidepressant medication and less than one-half of those taking multiple antidepressants have seen a mental health professional in the past year. This data brief discusses all antidepressants taken, regardless of the reason for use. Pharmacoepidemiol Drug Saf 16(5):560-70.
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