Inversion (discrete mathematics): Difference between revisions

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Let <math>\pi</math> be a [[permutation]].

There is an '''inversion''' of <math>\pi</math> between <math>i</math> and <math>j</math> if <math>i < j</math> and <math>\pi(i) > \pi(j)</math>. The inversion is indicated by an ordered pair containing either the places <math>(i, j)</math>{{sfn|Aigner|2007|pp=27}}{{sfn|Comtet|1974|pp=237}} or the elements <math>\bigl(\pi(i), \pi(j)\bigr)</math>.{{sfn|Knuth|1973|pp=11}}{{sfn|Pemmaraju|Skiena|2003|pp=69}}{{sfn|Vitter|Flajolet|1990|pp=459}}. Which value of an inversion's ordered pair is smaller indicates whether place-based notation (first value smaller) or element-based notation (second value smaller) is being used.

The '''[[#Example:_All_permutations_of_four_elements|inversion set''']] is the set of all inversions. A permutation's inversion set according to theusing place-based notation is thatthe ofsame as the [[Permutation#Product and inverseDefinition|inverse permutation's]] inversion set using element-based notation with the two components of each ordered pair exchanged. Likewise, a permutation's inversion set accordingusing toelement-based notation is the elementsame as the inverse permutation's inversion set using place-based notation, andwith vicethe versa,two components of each ordered pair exchanged.{{sfn|Gratzer|2016|pp=221}} just with the elements of the pairs exchanged.

InversionInversions isare usually defined for permutations, but may also be defined for sequences:<br>Let <math>S</math> be a [[sequence]] (or [[multiset]] permutation{{sfn|Bóna|2012|pp=57}}). If <math>i < j</math> and <math>S(i) > S(j)</math>, either the pair of places <math>(i, j)</math>{{sfn|Bóna|2012|pp=57}}{{sfn|Cormen|Leiserson|Rivest|Stein|2001|pp=39}} or the pair of elements <math>\bigl(S(i), S(j)\bigr)</math>{{sfn|Barth|Mutzel|2004|pp=183}} is called an inversion of <math>S</math>.

For sequences, inversions according to the element-based definition are not unique, because different pairs of places may have the same pair of values.

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===Inversion number===

The '''inversion number''' <math>\mathtt{inv}(X)</math>{{sfn|Mannila|1985}} of a sequence <math>X=\langle x_1,\dots,x_n\rangle</math>, is the [[cardinality]] of the inversion set. It is a common [[measure of presortedness|measures ofsortedness (pre-sometimes called presortedness)sortedness]] of a permutation{{sfn|Vitter|Flajolet|1990|pp=459}} or sequence.{{sfn|Barth|Mutzel|2004|pp=183}} The inversion number is between 0 and <math>\frac{n(n-1)}2</math> inclusive. A permutation and its inverse have the same inversion number.

For example <math>\mathtt{inv}(\langle1,2,\dots, n\rangle)=0</math> since the sequence is ordered. Also, when <math>n = 2m</math> is even, <math>\mathtt{inv}(\langle nm+1,nm+2,\dots,2n2m,1,2,\dots, nm\rangle)=nm^2</math> as(because each pair <math>(1\le i\le nm < j\le 2n2m)</math> is an inversion). This last example shows that a set that is intuitively "nearly" sorted" can still have a quadratic number of inversions.

The inversion number is the number of crossings in the arrow diagram of the permutation,{{sfn|Gratzer|2016|pp=221}} the permutation's [[Kendall tau distance]] from the identity permutation, and the sum of each of the inversion related vectors defined below.

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, the Spearman footrule (sum of distances of each element from its sorted position), 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'')}}.{{sfn|Kleinberg|Tardos|2005|pp=225}}

===Inversion related vectors===

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:<math>r(i) ~~=~~ \# \{ k \mid k > i ~\land~ \pi(k) < \pi(i) \}</math>

Both <math>v</math> and <math>r</math> can be found with the help of a '''[[Rothe diagram''']], which is a [[permutation matrix]] with the 1s represented by dots, and an inversion (often represented by a cross) in every position that has a dot to the right and below it. <math>r(i)</math> is the sum of inversions in row <math>i</math> of the Rothe diagram, while <math>v(i)</math> is the sum of inversions in column <math>i</math>. The permutation matrix of the inverse is the transpose, therefore <math>v</math> of a permutation is <math>r</math> of its inverse, and vice versa.

==Example: All permutations of four elements==

[[File:2-element subsets of 4 elements; array, hexagonal.svg|thumb|The six possible inversions of a 4-element permutation]]

The following sortable table shows the 24 permutations of four elements (in the <math>\pi</math> column) with their place-based inversion sets (in the p-b column), inversion related vectors (in the <math>v</math>, <math>l</math>, and <math>r</math> columns), and inversion numbers (in the # column). (The small columns with smaller print and no heading are reflections of the columns next to them, and can be used to sort them in [[colexicographic order]].)

It can be seen that <math>v</math> and <math>l</math> always have the same digits, and that <math>l</math> and <math>r</math> are both related to the place-based inversion set. The nontrivial elements of <math>l</math> are the sums of the descending diagonals of the shown triangle, and those of <math>r</math> are the sums of the ascending diagonals. (Pairs in descending diagonals have the right components 2, 3, 4 in common, while pairs in ascending diagonals have the left components 1, 2, 3 in common.)

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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]].

The [[Hasse diagram]] of the inversion sets ordered by the [[subset]] relation forms the [[skeleton (topology)|skeleton]] of a [[permutohedron]].

If a permutation is assigned to each inversion set using the place-based definition, the resulting order of permutations is that of the permutohedron, where an edge corresponds to the swapping of two elements with consecutive values. This is the weak order of permutations. The identity is its minimum, and the permutation formed by reversing the identity is its maximum.

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== See also ==

{{wikiversity|Inversion (discrete mathematics)}}

{{Commons category|Inversion (discrete mathematics)}}

* [[Factorial number system]]

* [[Permutation graph]]

* [[Permutation group#Transpositions, simple transpositions, inversions and sorting|Transpositions, simple transpositions, inversions and sorting]]

* [[Damerau–Levenshtein distance]]

* [[Parity of a permutation]]

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|url = https://archive.org/details/Comtet_Louis_-_Advanced_Coatorics

| chapter = 6.4 Inversions of a permutation of [n]

| publisher = D. Reidel Pub. Co | ___location = Dordrecht, Boston | year = 1974 | isbn = 9027704414 }}

* {{cite book

| first1=Thomas H. |last1=Cormen |authorlink1=Thomas H. Cormen

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|title=Algorithm Design

|year=2005

|publisher=Pearson/Addison-Wesley

|isbn=0-321-29535-8 }}

* {{cite book

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=== Further reading ===

{{refbegin}}

* {{cite journal|journal=Journal of Integer Sequences|volume=4|year=2001|title=Permutations with Inversions|first=Barbara H.|last=Margolius|page=24|bibcode=2001JIntS...4...24M}}

{{refend}}