Inversion (discrete mathematics)
Inversion is a concept in discrete mathematics to measure how much a sequence is out of its natural order.
(An inversion of a permutation is not to be confused with the inverse of a permutation. Also not with point reflection.)
Definitions
[edit  edit source]A permutation has an inversion for each pair of elements that are out of their natural order, i.e. when but .
It may be identified by the pair of places or by the pair of elements .
This article favours the former convention (although the latter seems to be more common).
The inversion set of a permutation is the set of all its inversions.
For an element permutation the number of potential elements is the triangular number .
The elementbased inversion set is essentially the placebased inversion set of the inverse permutation, just with the elements of the pairs exchanged.
For sequences the elementbased inversion set has to be defined as a multiset, as many pairs of places can have the same pair of elements.
The permutation graph has the elements of the permutation as vertices and an edge between elements that are out of order,
i.e. an edge is an inversion according to the elementbased definition: is an edge iff .
It is more common to write that is an edge iff . The latter is often written as .
The permutation graphs of inverse permutations are isomorphic. That of can be generated from that of by changing each vertex to .
It seems that the permutation graph is only defined with placebased inversions. If it were generalized to apply to sequences, it would thus have to be defined as a multigraph.
The inversion number ( A034968 ) is the cardinality of the inversion set;
thus also the digit sum of the inversion related vectors;
and also the number of crossings in the arrow diagram of the permutation.
The parity of a permutation is the parity of its inversion number.
The sign of a permutation (the determinant of its matrix) corresponds to the parity: Even permutations have sign 1, odd permutations sign −1.
Inversion related vectors
[edit  edit source]There are four ways to condense the inversions of a permutation into a vector that uniquely determines it. Three of them are in use. (See sources below).
This article uses the terms left inversion count (l) and right inversion count (r) like Gnedin (similar to Calude and Deutsch), and the term inversion vector (v) like Wolfram.
The left and right inversion counts have the feature that interpreted as factorial numbers they determine the permutation's reverse colexicographic and lexicographic index.
The inversion vector does not appear to have a similar advantage, but it is widely used anyway.
Left inversion count l:
With the placebased definition is the number of inversions whose bigger (right) component is .
 is the number of elements in greater than before .
Right inversion count r, often called Lehmer code:
With the placebased definition is the number of inversions whose smaller (left) component is .
 is the number of elements in smaller than after .
Inversion vector v:
With the elementbased definition is the number of inversions whose smaller (right) component is .
 is the number of elements in greater than before .
A different way to say the same thing may be more intuitive:
 is the number of elements in greater than before .
The latter definition would also work for a sequence, which does not have an inverse.
Python functions  

def left_inversion_count(p):
n = len(p)
v = [0] * n
for i in range(n):
count = 0
for k in range(i):
if p[k] > p[i]:
count += 1
v[i] = count
return v

def right_inversion_count(p):
n = len(p)
v = [0] * n
for i in range(n):
count = 0
for k in range(i, n):
if p[k] < p[i]:
count += 1
v[i] = count
return v

def inversion_vector(p_raw):
if min(p_raw) == 1: # assuming 1based permutation
p = [entry1 for entry in p_raw]
else: # assuming 0based permutation
p = p_raw
n = len(p)
v = [0] * n
for i in range(n):
count = 0
for k in range(i):
if p[k] > p[i]:
count += 1
v[p[i]] = count
return v

Relationship between l and v:

(vector permuted by is , compare Permutation notation)  

The first digit of l and the last digit of v are always 0, and can be omitted.
When these vectors are permuted into each other, the omissible 0 from one does not necessarily become the omissible 0 in the other.
Relationship between r and v:
Both r and v can be found with the help of a Rothe diagram,
which is a permutation matrix with the 1s represented by dots, and an inversion in every position that has a dot to the right and below it.
is the sum of inversions in row of the Rothe diagram, while is the sum of inversions in column .
The permutation matrix of the inverse is the transpose. Therefore v of a permutation is r of its inverse, and vice versa.
Relationship between r and l:
This is proven in Gnedin & Olshanski 2012, and mentioned as R[i]+i=p[i]+L[i]
in A138159.
Permutations that are equal up to fixed points can be seen as equal. Equivalently, inversion related vectors that are equal up to trailing 0s can be seen as equal.
While for r and v the omissible 0 on the right is part of the trailing 0s, for l the omissible 0 on the left is separate from them.
Example: Permutations of six elements
[edit  edit source]The following images show two inverse permutations, their 9 inversions and the corresponding vectors.
l = (0, 0, 1, 0, 3, 5) 
Formulas  

l = (0, 1, 1, 2, 3, 2) 
Mathematica omits the 0 at the end of the inversion vector. Compare p1 and p2 in Wolfram Alpha.
I: p1 = {2,5,4,6,3,1} I: p2 = {6,1,5,3,2,4} I: ToInversionVector[p1] O: {5,0,3,1,0} I: ToInversionVector[p2] O: {1,3,2,2,1} I: ToOrderedPairs[PermutationGraph[p1]] O: {{2,1},{3,1},{4,1},{5,1},{6,1},{4,3},{5,3},{6,3},{5,4},{1,2},{1,3},{1,4},{1,5},{1,6},{3,4},{3,5},{3,6},{4,5}} I: ToOrderedPairs[PermutationGraph[p2]] O: {{6,1},{3,2},{5,2},{6,2},{5,3},{6,3},{5,4},{6,4},{6,5},{1,6},{2,3},{2,5},{2,6},{3,5},{3,6},{4,5},{4,6},{5,6}}
Example: All permutations of four elements
[edit  edit source]The following sortable table shows the 24 permutations of four elements with their placebased inversion sets, inversion related vectors and inversion numbers.
It is shown twice, so different orders can be compared to each other. (The Python script used to create it is shown here.)
The columns with small text are the reflections of the main columns. Sorting by them gives the colexicographic order of the corresponding main column.
Inversion vector v and left inversion count l are shown next to each other, because they have the same digits.
Left and right inversion counts are related to the placebased inversion sets shown between them:
The nontrivial elements of l are the sums of the descending diagonals of the triangle, and those of r are the sums of the of the ascending diagonals.
The default order of the table is the reverse colex order by , which is the same as the colex order by l. (This header cell is highlighted by a darker gray.)
Lex order by is the same as lex order by r.
Ordering one table by r and the other one by v brings inverse permutations next to each other, i.e. those whose matrices are reflected at the main diagonal.
Ordering one table by reflected l and the other one by v brings permutations next to each other whose permutation matrices are reflected at the antidiagonal.
Ordering one table by reflected l and the other one by r brings permutations next to each other whose permutation matrices are rotated by 180°.
(It can be seen, that these permutations' inversion sets are symmetric to each other, which corresponds to l and r being symmetric to each other.)
Note that the set of all r, the set of all v and the set of all reflected l for the same symmetric group are equal.
So sorting by two columns that have the omissible 0 at the same end makes these columns equal.


(A more detailled version of this table, including elementbased inversion sets and the unused fourth vector, can be found here.)
Weak order of permutations
[edit  edit source]The Hasse diagram below in the middle shows the 24 inversion sets ordered by the subset relation.
If a permutation is assigned to each inversion set using the placebased definition, the resulting order of permutations is that of the permutohedron,
where is an edge iff , i.e. when only two elements with consecutive values are swapped.
(In the central diagram it can be seen on which positions the swapped elements are, by looking which digit changes for the edge.)
The bitwise ordering of the left and right inversion counts of these permutations gives the same order (because both are diagonal sums of the inversion set triangles).
Schlegel diagrams of the permutohedra above  


If a permutation were assigned to each inversion set using the elementbased definition, the resulting order of permutations would be that of a Cayley graph,
where is an edge iff , i.e. when only two elements on consecutive places are swapped.
This Cayley graph of the symmetric group is similar to its permutohedron, but with each permutation replaced by its inverse.
Cayley graph  


For sequences inversion sets and the related vectors are not unique. Their placebased inversion sets are multisets, and their inversion vectors can have bigger digits than would be allowed in a factorial number.


Similar permutations have similar inversion sets and corresponding vectors.
Some can be ordered in arrays, like the one on the right, corresponding to the number triangle A211367.
Sources
[edit  edit source]Books
[edit  edit source]Cramer 1750
[edit  edit source]Cramer defines the inversion under the name dérangement to calculate the sign of a permutation.
For the permutation 3 1 2 the dérangements are given as 3 before 1 and 3 before 2.
Cramer, Gabriel: Introduction à l'analyse des lignes courbes algébriques. Genève : chez les Frères Cramer & Cl. Philibert 1750 — Appendice (p. 658)
Chabert (1999). A history of algorithms : from the pebble to the microchip. Berlin New York: Springer. ISBN 9783540633693. — 9.1 Cramers rule (p. 286)
Rothe 1800
[edit  edit source]Rothe essentially defines the right inversion count r under the name Stellenexponenten — but with each place bigger by 1.
 is the position of among the with in numerical order. (p. 163)
Strangely he counts from 1 rather than 0, which later leads to avoidable additions and subtractions by 1:
For the permutation the Stellenexponenten are given as .
Muir translates Stellenexponent as placeindex and calls it “an illadvised and purposeless modification of Cramers idea of a ‘derangement’”. (The Theory of the Determinant..., p. 55)
"Ueber Permutationen, in Beziehung auf die Stellen ihrer Elemente. Anwendung der daraus abgeleiteten Satze auf das Eliminationsproblem". In Hindenburg, Carl, ed., Sammlung CombinatorischAnalytischer Abhandlungen, pp. 263–305, Bey G. Fleischer dem jüngern, 1800.
Laisant 1888
[edit  edit source]After defining the factorial number system Laisant defines the right inversion count r under the name signe figuratif.
For the permutation the signe figuratif is given as .
Laisant, CharlesAnge (1888), "Sur la numération factorielle, application aux permutations", Bulletin de la Société Mathématique de France (in French), 16: 176–183
Aigner 2007
[edit  edit source]pb perm 
Now we look at permutations of in word form .
The pair is called an inversion if but .A moment's thought shows that is an inversion if and only if the edges and cross in the diagram.
Hence the inversion number equals the number of crossings.
Aigner, Martin (2007). A course in enumeration. Berlin New York: Springer. ISBN 3642072534. — Word Representation (p. 27 ff)
Comtet 1974
[edit  edit source]pb perm 
An inversion of a permutation is a pair such that and . In this case we say that has an inversion in .
He uses the same definition for a nonbijective map in exercise 21 on permutations (p. 266).
Comtet, Louis (1974). Advanced combinatorics; the art of finite and infinite expansions. Dordrecht,Boston: D. Reidel Pub. Co. ISBN 9027704414. — 6.4 Inversions of a permutation of [n] (p. 237)
Cormen et al. 2001
[edit  edit source]pb seq 
Let be an array of n distinct numbers. If and , then the pair is called an inversion of .
Cormen, Thomas (2001). Introduction to algorithms. Cambridge, Mass: MIT Press. ISBN 0262531968. — 24 Inversions (p. 41) and 5.25 (p. 122)
Knuth 1973
[edit  edit source]eb perm v as inversion table 
If and , the pair is called an inversion of the permutation; for example the permutation 3 1 4 2 has three inversions: (3, 1), (3, 2) and (4, 2).
The inversion table of the permutation is obtained by letting be the number of elements to the left of that are greater than .
In other words, is the number of inversions whose second component is .
Knuth, Donald (1973). The art of computer programming. Reading, Mass: AddisonWesley Pub. Co. ISBN 0201896850. — 5.1.1 Inversions (p. 11)
Pemmaraju & Skiena 2003
[edit  edit source]eb perm v as inversion vector 
A pair of elements in a permutation represents an inversion if and .
An inversion is a pair of elements that are out of order [...]For any permutation , we can define an inversion vector as follows. For each integer , , the th element of is the number of elements in greater than to the left of .
Pemmaraju, Sriram (2003). Computational discrete mathematics : combinatorics and graph theory with Mathematica. Cambridge, U.K. New York: Cambridge University Press. ISBN 9780521806862. — 2.2 Inversions and Inversion Vectors (p. 69)
Vitter & Flajolet 1990
[edit  edit source]eb perm v as inversion table 
An inversion in permutation is an “out of order” pair of elements, in which but .
The number of inversions is thus a measure of the amount of disorder in a permutation.The inversion table of the permutation is the ordered sequence
 , where .
Leeuwen, J (1990). Handbook of theoretical computer science. Amsterdam New York Cambridge, Mass: Elsevier MIT Press. ISBN 9780444880710. — 3.1 Inversions (p. 459)
Grätzer 2016
[edit  edit source]eb perm 
The authors define the usual strict linear ordering on as .
Let be a permutation on . An inversion of is an ordered pair satisfying .
The inversion set of is then defined as .
The inversion set of is given as .
In the footnote it is mentioned that other authors often use the definition as an ordered pair such that .
TAoCP is given as an example — despite the fact that Knuth uses essentially the same definition (see above).
Gratzer, George (2016). Lattice theory. special topics and applications. Cham, Switzerland: Birkhäuser. ISBN 331944235X. — 72 Basic objects (p. 221)
Joshi 1989
[edit  edit source]eb seq r as inversion table or inversion vector 
Let be a sequence of real numbers.
By an inversion in , we mean a pair such that but .
For each , let = the number of inversions whose first entry is , i.e.
 .
Then the sequence is called the inversion table or inversion vector of . [...]
The permutation has as its inversion table.
Joshi, K. D. (1989). Foundations of discrete mathematics. New York New Delhi, India: Wiley Wiley Eastern Ltd. ISBN 8122401201. — Definition 3.12 (p. 188)
Bóna 2012
[edit  edit source]eb perm 
Bóna first uses the definition with elements:
Let be a permutation. We say that is an inversion of if but .
Permutation 31524 has four inversions, namely (3, 1), (3, 2), (5, 2), and (5,4).
2.1 Inversions (p. 43)
pb seq 
But for multisets he uses places instead:
An inversion of a permutation of a multiset is defined just as it was for permutations of sets, that is, is a inversion if , but ·
The multisetpermutation 1322 has two inversions, (2, 3), and (2, 4).
2.2 Inversions in Permutations of Multisets (p. 57)
The definition of noninversions also uses places:
[...] look at all noninversions of a generic permutation ; that is, pairs so that and .
4.4.1.1 An exponential upper bound (p. 141)
Bóna, Miklós (2012). Combinatorics of permutations. Boca Raton, FL: CRC Press. ISBN 1439850518.
Calude et al. 2003
[edit  edit source]l as leftinversion vector r as rightinversion vector 
For permutations, an inversion vector is used as an auxiliary array because it fills in the holes made by the elements of the prefix in its defining sequence.
For the permutation the rightinversion vector is defined by the rule that is the number of to the right of but smaller than ;
in the leftinversion vector, is the number of to the left of but larger than .
Calude, Cristian (2003). Discrete mathematics and theoretical computer science : 4th international conference, DMTCS 2003, Dijon, France, July 712, 2003 : proceedings. Berlin New York: Springer. ISBN 3540405054. — Generating Gray codes... (p. 81)
Lothaire 2002
[edit  edit source]pb perm l as Lehmer encoding 
If is [a permutation of ] an inversion of is a pair such that and .
11.1. Prelimiaries (p. 367)
Let be a permutation of . For , let be the number of indices such that . The word will be called the Lehmer encoding of .
To the permutation corresponds the Lehmer encoding .
11.4. Inversions of permutations with a given shape (p. 372)
Lothaire, M (2002). Algebraic combinatorics on words. Cambridge New York: Cambridge University Press. ISBN 0521812208.
Papers
[edit  edit source]Vajnovszki 2011
[edit  edit source]pb perm l as Lehmer code 
The pair is an inversion of if but . [...]
An integer sequence is said to be subexcedent if for , and the set of lenth subexcedent sequences is denoted by [...].
The Lehmer code [5] is a bijection code : which maps each permutation to a subexcedent sequence
where, for all , is the number of inversions in (or equivalently, the number of entries in larger than and on its left).
[5] D. H. Lehmer, Teaching combinatorial tricks to a computer, in Proc. Sympos. Appl. Math., 10 (1960), Amer. Math. Soc., 179193.
Vajnovszki. A new Euler–Mahonian constructive bijection — pp. 12
Gnedin & Olshanski 2012
[edit  edit source]pb perm l as left inversion count r as right inversion count 
For , a pair of positions is an inversion in if and .
If is an inversion, we say that it is a left inversion for , and a right inversion for .
Introduce the counts of left and right inversions,
 ,
respectively (of course, for the general these quantities may be infinite).
4. A construction from independent geometric variables (p. 624)
The authors proof the following formula for balanced permutations, which include finite permutations:
Lemma 4.6 (p. 627)
Gnedin; Olshanski. The twosided infinite extension of the Mallows model for random permutations
Deutsch et al. 2008
[edit  edit source]pb perm r as right inversion vector 
In a permutation , an inversion is a pair such that . [...]
If is the number of with , then is called the right inversion vector of [...].
Deutsch; Pergola; Pinzani. Six bijections between deco polyominoes and permutations — p. 5
Barth & Mutzel 2004
[edit  edit source]eb seq 
In a sequence of pairwise comparable elements , a pair is called an inversion if and .
A crucial point of the article is illustrated by the images on the right: A bilayer graph that does not describe a map corresponds to a map that has the edges (green) as its domain and one layer of the vertices (blue) as its codomain. The number of crossings in the bilayer graph can be calculated as the number of inversions of the map.
In this example the map is .
, and , so with the placebased definition the sequence would have the inversions and .
With the elementbased definition, chosen by the authors, one could say that the sequence has the inversion twice.
Barth, Wilhelm; Mutzel, Petra (2004). "Simple and Efficient Bilayer Cross Counting". Journal of Graph Algorithms and Applications 8 (2): 179–194. doi:10.7155/jgaa.00088.
— 2 Bilayer Cross Counts and Inversion Numbers (p. 183)