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
A permutation $$\pi$$ has an inversion for each pair of elements that are out of their natural order, i.e. when $$i < j$$ but $$\pi(i) > \pi(j)$$. It may be identified by the pair of places $$(i, j)$$ or by the pair of elements $$\bigl(\pi(i), \pi(j)\bigr)$$. 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 $$n$$-element permutation the number of potential elements is the triangular number $$\scriptstyle { n \choose 2}$$.

The element-based inversion set is essentially the place-based inversion set of the inverse permutation, just with the elements of the pairs exchanged.

For sequences the element-based 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 element-based definition: $$\{\pi(i), \pi(j)\}$$ is an edge iff $$i < j \land  \pi(i) > \pi(j)$$. It is more common to write that $$\{i, j\}$$ is an edge iff $$i < j \land  \pi^{-1}(i) > \pi^{-1}(j)$$. The latter is often written as $$(i - j) \cdot \bigl(\pi^{-1}(i) - \pi^{-1}(j)\bigr) < 0$$.

The permutation graphs of inverse permutations are isomorphic. That of $$\pi^{-1}$$ can be generated from that of $$\pi$$ by changing each vertex $$i$$ to $$\pi(i)$$.

It seems that the permutation graph is only defined with place-based inversions. If it were generalized to apply to sequences, it would thus have to be defined as a multigraph.

The inversion number 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
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 and right inversion count  like Gnedin (similar to Calude and Deutsch), and the term inversion vector  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 : With the place-based definition $$l(i)$$ is the number of inversions whose bigger (right) component is $$i$$.
 * $$l(i)$$ is the number of elements in $$\pi$$ greater than $$\pi(i)$$ before $$\pi(i)$$.
 * $$l(i) = \# \left\{ k \mid k < i ~\land~ \pi(k) > \pi(i) \right\}$$

Right inversion count, often called : With the place-based definition $$r(i)$$ is the number of inversions whose smaller (left) component is $$i$$.
 * $$r(i)$$ is the number of elements in $$\pi$$ smaller than $$\pi(i)$$ after $$\pi(i)$$.
 * $$r(i) = \# \{ k \mid k > i ~\land~ \pi(k) < \pi(i) \}$$

Inversion vector : With the element-based definition $$v(i)$$ is the number of inversions whose smaller (right) component is $$i$$.
 * $$v(i)$$ is the number of elements in $$\pi$$ greater than $$i$$ before $$i$$.
 * $$v(i) = \# \{ k \mid k > i ~\land~ \pi^{-1}(k) < \pi^{-1}(i) \}$$

A different way to say the same thing may be more intuitive:
 * $$v\bigl(\pi(i)\bigr)$$ is the number of elements in $$\pi$$ greater than $$\pi(i)$$ before $$\pi(i)$$.
 * $$v\bigl(\pi(i)\bigr) = \# \{ k \mid k < i ~\land~ \pi(k) > \pi(i) \} ~= l(i)$$

The latter definition would also work for a sequence, which does not have an inverse.

Relationship between and :

The first digit of and the last digit of  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 and : Both and  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. $$r(i)$$ is the sum of inversions in row $$i$$ of the Rothe diagram, while $$v(i)$$ is the sum of inversions in column $$i$$. The permutation matrix of the inverse is the transpose. Therefore of a permutation is  of its inverse, and vice versa.

Relationship between and :
 * $$\pi(i) = i + r(i) - l(i)$$

This is proven in Gnedin & Olshanski 2012, and mentioned as  in.

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 and  the omissible 0 on the right is part of the trailing 0s, for  the omissible 0 on the left is separate from them.

Example: Permutations of six elements
The following images show two inverse permutations, their 9 inversions and the corresponding vectors.

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


The following sortable table shows the 24 permutations of four elements with their place-based 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 and left inversion count  are shown next to each other, because they have the same digits.

Left and right inversion counts are related to the place-based inversion sets shown between them: The nontrivial elements of are the sums of the descending diagonals of the triangle, and those of  are the sums of the of the ascending diagonals.

The default order of the table is the reverse colex order by $$\pi$$, which is the same as the colex order by. (This header cell is highlighted by a darker gray.)

Lex order by $$\pi$$ is the same as lex order by.

Ordering one table by and the other one by  brings inverse permutations next to each other, i.e. those whose matrices are reflected at the main diagonal.

Ordering one table by reflected and the other one by  brings permutations next to each other whose permutation matrices are reflected at the antidiagonal.

Ordering one table by reflected and the other one by  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 and  being symmetric to each other.)

Note that the set of all, the set of all and the set of all reflected  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 element-based inversion sets and the unused fourth vector, can be found here.)

Weak order of permutations
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 place-based definition, the resulting order of permutations is that of the permutohedron, where $$(\pi, \sigma)$$ is an edge iff $$i < j \land  \pi(i)+1 = \pi(j)  \land  (ij) \cdot \pi = \sigma$$, 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 $$\le$$ 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).

If a permutation were assigned to each inversion set using the element-based definition, the resulting order of permutations would be that of a Cayley graph, where $$(\pi, \sigma)$$ is an edge iff $$i+1 = j \land  \pi(i) < \pi(j)  \land  (ij) \cdot \pi = \sigma$$, 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.

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

Arrays of permutations
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. 

Cramer 1750
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)

— 9.1 Cramers rule (p. 286)

Rothe 1800
Rothe essentially defines the right inversion count under the name Stellenexponenten — but with each place bigger by 1.


 * $$s_i$$ is the position of $$\pi_i$$ among the $$\pi_k$$ with $$k \ge i$$ in numerical order. (p. 163)

Strangely he counts from 1 rather than 0, which later leads to avoidable additions and subtractions by 1:
 * $$s_i + 1 = \# \{ \pi_k \mid k>i \land \pi_k<\pi_i \}$$ (p. 165)
 * Each permutation has sign $$+$$ when the sum of the $$s_i - 1$$ is even, and sign $$-$$ when it is odd. (p. 266)

For the permutation $$6~4~3~9~8~10~1~7~2~5$$ the Stellenexponenten are given as $$6~4~3~6~5~5~1~3~1~1$$.

Muir translates Stellenexponent as place-index and calls it “an ill-advised 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 Combinatorisch-Analytischer Abhandlungen, pp. 263–305, Bey G. Fleischer dem jüngern, 1800.

Laisant 1888
After defining the factorial number system Laisant defines the right inversion count under the name signe figuratif.

For the permutation $$4~3~6~5~1~2$$ the signe figuratif is given as $$(3~2~3~2~0)$$.

Aigner 2007
Now we look at permutations of $$\{1, 2, ..., n\}$$ in word form $$\sigma = a_1, a_2 ... a_n, a_i = \sigma(i)$$. The pair $$\{i, j\}$$ is called an inversion if $$i < j$$ but $$a_i > a_j$$.

A moment's thought shows that $$\{i, j\}$$ is an inversion if and only if the edges $$ia_i$$ and $$ja_j$$ cross in the diagram. Hence the inversion number $$inv(\sigma)$$ equals the number of crossings.

— Word Representation (p. 27 ff)

Comtet 1974
An inversion of a permutation $$\sigma \in \mathfrak{S}[n]$$ is a pair $$(i, j)$$ such that $$1 \le i < j \le n$$ and $$\sigma(i) > \sigma(j)$$. In this case we say that $$\sigma$$ has an inversion in $$(i, j)$$.

He uses the same definition for a non-bijective map in exercise 21 on permutations (p. 266).

— 6.4 Inversions of a permutation of [n] (p. 237)

Cormen et al. 2001
Let $$A[1..n]$$ be an array of n distinct numbers. If $$i < j$$ and $$A[i] > A[j]$$, then the pair $$(i, j)$$ is called an inversion of $$A$$.

— 2-4 Inversions (p. 41) and 5.2-5 (p. 122)

Knuth 1973
If $$ia_j$$, the pair $$(a_i, a_j)$$ 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 $$b_1 b_2 ... b_n$$ of the permutation $$a_1 a_2 ... a_n$$ is obtained by letting $$b_j$$ be the number of elements to the left of $$j$$ that are greater than $$j$$. In other words, $$b_j$$ is the number of inversions whose second component is $$j$$.

— 5.1.1 Inversions (p. 11)

Pemmaraju & Skiena 2003
A pair of elements $$(\pi(i), \pi(j))$$ in a permutation $$\pi$$ represents an inversion if $$i > j$$ and $$\pi(i) < \pi(j)$$. An inversion is a pair of elements that are out of order

For any $$n$$-permutation $$\pi$$, we can define an inversion vector $$v$$ as follows. For each integer $$i$$, $$1 \le i \le n-1$$, the $$i$$th element of $$v$$ is the number of elements in $$\pi$$ greater than $$i$$ to the left of $$i$$.

— 2.2 Inversions and Inversion Vectors (p. 69)

Vitter & Flajolet 1990
An inversion in permutation $$\sigma$$ is an “out of order” pair $$(\sigma_k, \sigma_j)$$ of elements, in which $$k \sigma_j$$. The number of inversions is thus a measure of the amount of disorder in a permutation.

The inversion table of the permutation $$\sigma = \sigma_1 \sigma_2 \dots \sigma_n$$ is the ordered sequence
 * $$b_1, b_2, \dots, b_n$$, where $$b_k = \Big| \{ 1 \le j <\sigma_k^-1 | \sigma_j > k \} \Big|$$.

— 3.1 Inversions (p. 459)

Grätzer 2016
The authors define the usual strict linear ordering $$1 < 2 < \cdots < i < i+1 < \cdots < n$$ on $$[n]$$ as $$\mathcal{I}_n = \{ (i, j) \in [n] \times [n]  |  i < j  \}$$.

Let $$\sigma$$ be a permutation on $$[n]$$. An inversion of $$\sigma$$ is an ordered pair $$(i, j) \in \mathcal{I}_n$$ satisfying $$\sigma^{-1}(i) > \sigma^{-1}(j)$$. The inversion set of $$\sigma$$ is then defined as $$\mathrm{inv}(\sigma) = \{ (i, j) \in \mathcal{I}_n  |  \sigma^{-1}(i) > \sigma^{-1}(j)  \}$$.

The inversion set of $$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 5 & 4 & 1 & 2 \end{pmatrix}$$ is given as $$\{(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (4, 5)\}$$.

In the footnote it is mentioned that other authors often use the definition as an ordered pair $$(i, j) \in \mathcal{I}_n$$ such that $$\sigma(i) > \sigma(j)$$. TAoCP is given as an example — despite the fact that Knuth uses essentially the same definition (see above).

— 7-2 Basic objects (p. 221)

Joshi 1989
Let $$x = x_1 x_2 \dots x_n$$ be a sequence of real numbers. By an inversion in $$x$$, we mean a pair $$(x_i, x_j)$$ such that $$ix_j$$. For each $$i$$, let $$d_i$$ = the number of inversions whose first entry is $$x_i$$, i.e.
 * $$d_i = \big| \{ x_j : i < j, x_i > x_j \} \big|$$.

Then the sequence $$(d_1 d_2 \dots d_n)$$ is called the inversion table or inversion vector of $$x$$. The permutation $$2~4~3~5~1$$ has $$(1, 2, 1, 1, 0)$$ as its inversion table.

— Definition 3.12 (p. 188)

Bóna 2012
Bóna first uses the definition with elements: Let $$p = p_1 p_2 \cdots p_n$$ be a permutation. We say that $$(p_i, p_j)$$ is an inversion of $$p$$ if $$i < j$$ but $$p_i > p_j$$. Permutation 31524 has four inversions, namely (3, 1), (3, 2), (5, 2), and (5,4). 2.1 Inversions (p. 43)

But for multisets he uses places instead: An inversion of a permutation $$p = p_1 p_2 \cdots p_n$$ of a multiset is defined just as it was for permutations of sets, that is, $$(i, j)$$ is a inversion if $$i < j$$, but $$p_i > p_j$$· The multiset-permutation 1322 has two inversions, (2, 3), and (2, 4). 2.2 Inversions in Permutations of Multisets (p. 57)

The definition of non-inversions also uses places: look at all non-inversions of a generic permutation $$p = p_1 p_2 \cdots p_n$$; that is, pairs so that $$i < j$$ and $$p_i < p_j$$. 4.4.1.1 An exponential upper bound (p. 141)

Calude et al. 2003
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 $$(p[i], \dots, p[n])$$ the right-inversion vector $$(g[i], \dots, g[n-1])$$ is defined by the rule that $$g[i]$$ is the number of $$p[j]$$ to the right of but smaller than $$p[i]$$; in the left-inversion vector, $$g[i]$$ is the number of $$p[j]$$ to the left of but larger than $$p[i]$$.

— Generating Gray codes... (p. 81)

Lothaire 2002
If $$\sigma = \sigma(1) \sigma(2) \dots \sigma(k)$$ is [ a permutation of $$[k]$$ ] an inversion of $$\sigma$$ is a pair $$(i, j)$$ such that $$1 \le i < j \le k$$ and $$\sigma(i) > \sigma(j)$$. 11.1. Prelimiaries (p. 367)

Let $$\sigma$$ be a permutation of $$[n]$$. For $$1 \le i \le n$$, let $$l_i$$ be the number of indices $$j < i$$ such that $$\sigma(j) > \sigma(i)$$. The word $$l = l_1 l_2 \cdots l_n$$ will be called the Lehmer encoding of $$\sigma$$.

To the permutation $$\sigma = 6~8~4~5~9~3~1~2~7$$ corresponds the Lehmer encoding $$l = 0~0~2~2~0~5~6~6~2$$. 11.4. Inversions of permutations with a given shape (p. 372)

Vajnovszki 2011
The pair $$(i, j)$$ is an inversion of $$\pi \in \mathfrak{S}_n$$ if $$i < j$$ but $$\pi_i > \pi_j$$.

An integer sequence $$t_1 t_2 \dots t_n$$ is said to be subexcedent if $$0 \le t_i \le i - 1$$ for $$1 \le i \le n$$, and the set of lenth-$$n$$ subexcedent sequences is denoted by $$S_n$$. The Lehmer code [5] is a bijection code : $$\mathfrak{S}_n \rightarrow S_n$$ which maps each permutation $$\pi = \pi_1 \pi_2 \dots \pi_n$$ to a subexcedent sequence $$t_1 t_2 \dots t_n$$ where, for all $$i, ~ 1 \le i \le n$$, $$~ t_i$$ is the number of inversions $$(j, i)$$ in $$\pi$$ (or equivalently, the number of entries in $$\pi$$ larger than $$\pi_i$$ and on its left). [5]     D. H. Lehmer, Teaching combinatorial tricks to a computer, in Proc. Sympos. Appl. Math., 10 (1960), Amer. Math. Soc., 179-193. Vajnovszki. A new Euler–Mahonian constructive bijection — pp. 1-2

Gnedin & Olshanski 2012
For $$\sigma \in \mathfrak{S}$$, a pair of positions $$(i, j) \in \mathbb{Z} \times \mathbb{Z}$$ is an inversion in $$\sigma$$ if $$i < j$$ and $$\sigma(i) > \sigma(j)$$. If $$(i, j)$$ is an inversion, we say that it is a left inversion for $$j$$, and a right inversion for $$i$$. Introduce the counts of left and right inversions,
 * $$\mathcal{l}_i := \# \{j: j < i, \sigma(j) > \sigma(i) \}, ~ \mathcal{r}_i := \# \{j: j > i, \sigma(j) < \sigma(i) \}, ~ i \in \mathbb{Z}$$,

respectively (of course, for the general $$\sigma \in \mathfrak{S}$$ 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:
 * $$\sigma(i) = i + \mathcal{r}_i - \mathcal{l}_i, ~ i \in \mathbb{Z}$$

Lemma 4.6 (p. 627)

Gnedin; Olshanski. The two-sided infinite extension of the Mallows model for random permutations

Deutsch et al. 2008
In a permutation $$\pi = \pi_1 \pi_2 \dots \pi_n$$, an inversion is a pair $$i < j$$ such that $$\pi_i>\pi_j$$. If $$c_i$$ is the number of $$j > i$$ with $$\pi_j < \pi_i$$, then $$(c_1, c_2, \dots, c_n)$$ is called the right inversion vector of $$\pi$$.

Deutsch; Pergola; Pinzani. Six bijections between deco polyominoes and permutations — p. 5

Barth & Mutzel 2004
In a sequence $$\pi = \langle a_0, a_1, \dots, a_{t-1} \rangle$$ of pairwise comparable elements $$a_i ~ (i = 0, 1, \dots , t-1)$$, a pair $$(a_i, a_j)$$ is called an inversion if $$i < j$$ and $$a_i > a_j$$.

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 $$\pi = \langle 0,1,2,0,3,4,0,2,3,2,4 \rangle$$.

$$\pi_2 = 2$$, $$\pi_3 = 0$$ and $$\pi_6 = 0$$, so with the place-based definition the sequence would have the inversions $$(2, 3)$$ and $$(2, 6)$$. With the element-based definition, chosen by the authors, one could say that the sequence has the inversion $$(2, 0)$$ twice.

— 2 Bilayer Cross Counts and Inversion Numbers (p. 183) 