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}}. As can be seen from this definition, which notation is being used can be determined by which value of the ordered pair is smaller (for place-based notation the first value is smaller, while for element-based notation the second value is smaller).

The [[Inversion_(discrete_mathematics)#Example:_All_permutations_of_four_elements|inversion set]] is the set of all inversions. A permutation's inversion set using place-based notation is the same as the [[Permutation#Definition|inverse permutation's]] inversion set using element-based notation with the two valuescomponents of each ordered pair exchanged. Likewise, a permutation's inversion set using element-based notation is the same as the inverse permutation's inversion set using place-based notation with the two valuescomponents of each ordered pair exchanged.{{sfn|Gratzer|2016|pp=221}}

Inversions are 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>.

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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 sortedness (sometimes called presortedness) 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, for even n and lettingwhen <math>mn =n/2 2m</math> is even, <math>\mathtt{inv}(\langle m+1,m+2,\dots,2m,1,2,\dots, m\rangle)=m^2</math> (because each pair <math>(1\le i\le m < j\le 2m)</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.

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[[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}}