Permutation patterns: basic definitions and notation
David Bevan (The Open University)
Permutations, containment and avoidance
A permutation is considered to be simply an arrangement of the numbers for some positive . The length of permutation is denoted , and or is used for the set of all permutations of length .
It is common to consider permutations graphically. Given a permutation , its plot consists of the the points in the Euclidean plane, for .
A permutation, or pattern, is said to be contained in, or to be a subpermutation of, another permutation , written or , if has a (not necessarily contiguous) subsequence whose terms are order isomorphic to (i.e. have the same relative ordering as) . From the graphical perspective, contains if the plot of results from erasing zero or more points from the plot of and then rescaling the axes appropriately. For example, contains because the subsequence (among others) is ordered in the same way as (see Figure 1).
If does not contain , we say that avoids . For example, avoids since it has no subsequence ordered in the same way as .
If is a list of distinct integers, the reduction or reduced form of , denoted , is the permutation obtained from by replacing its th smallest entry with . For example, we have . Thus, if there is a subsequence of such that .
Permutation structure
Given two permutations and with lengths and respectively, their direct sum is the permutation of length consisting of followed by a shifted copy of :
The skew sum is defined analogously. See Figure 2 for an illustration.
A permutation is called sum indecomposable if it cannot be expressed as the direct sum of two shorter permutations. A permutation is skew indecomposable if it cannot be expressed as the skew sum of two shorter permutations. Every permutation has a unique representation as the direct sum of a sequence of sum indecomposable permutations, and also as the skew sum of a sequence of skew indecomposable permutations. If a permutation is the direct sum of a sequence of decreasing permutations, then we say that the permutation is layered. See Figure 2 for an example.
An interval of a permutation corresponds to a contiguous sequence of indices such that the set of values is also contiguous. Graphically, an interval in a permutation is a square “box” that is not cut horizontally or vertically by any point not in it. Every permutation of length has intervals of lengths , and . If a permutation has no other intervals, then is said to be simple.
Given a permutation and nonempty permutations , the inflation of by , denoted , is the permutation obtained by replacing each entry of with an interval that is order isomorphic to . See Figure 3 for an illustration.
A simple permutation is thus a permutation that cannot be expressed as the inflation of a shorter permutation of length greater than . Conversely, every permutation except is the inflation of a unique simple permutation of length at least .
Sometimes we want to refer to the extremal points in a permutation. A value in a permutation is called a lefttoright maximum if it is larger than all the values to its left. Lefttoright minima, righttoleft maxima and righttoleft minima are defined analogously. See Figure 4 for an illustration.
Permutation statistics
An ascent in a permutation is a position such that . Similarly, a descent is a position such that . A pair of terms in a permutation such that and is called an inversion.
A permutation statistic is simply a map from the set of permutations to the nonnegative integers. Classical statistics include the following:

the number of descents

the number of inversions

the number of excedances

the major index^{1}^{1}1Named after Major Percy Alexander MacMahon., the sum of the positions of the descents
The statistics and are equidistributed. That is, for all and , the number of permutations of length with descents is the same as the number of permutations of length with excedances. Furthermore, and also have the same distribution. Any permutation statistic that is distributed like is said to be Eulerian, and a statistic that is distributed like is said to be Mahonian^{2}^{2}2See footnote 1..
Classical permutation classes
The subpermutation relation is a partial order on the set of all permutations. A classical permutation class, sometimes called a pattern class, is a set of permutations closed downwards (a downset) under this partial order. Thus, if is a member of a permutation class and is contained in , then it must be the case that is also a member of . From a graphical perspective, this means that erasing points from the plot of a permutation in always results in the plot of another permutation in when the axes are rescaled appropriately. It is common in the study of classical permutation classes to reserve the word “class” for sets of permutations closed under taking subpermutations.
It is natural to define a classical permutation class “negatively” by stating the minimal set of permutations that it avoids. This minimal forbidden set of patterns is known as the basis of the class. The class with basis is denoted , and or is used for the set of permutations of length in . As a trivial example, is the class of increasing permutations (i.e. the identity permutation of each length). As another simple example, the class of avoiders, , consists of those permutations that can be partitioned into two decreasing subsequences.
The basis of a permutation class is an antichain (a set of pairwise incomparable elements) under the containment order, and may be infinite. Classes for which the basis is finite are called finitely based, and those whose basis consists of a single permutation are called principal classes.
Nonclassical patterns
Permutation patterns have been generalised in a variety of ways.
A barred pattern is specified by a permutation with some entries barred (, for example). If is a barred pattern, let be the permutation obtained by removing all the bars in ( in the example), and let be the permutation that is order isomorphic to the nonbarred entries in ( in the example). An occurrence of barred pattern in a permutation is then an occurrence of in that is not part of an occurrence of in . Conversely, for to avoid , every occurrence in of must feature as part of an occurrence of .
A vincular or generalised pattern specifies adjacency conditions. Two different notations are used. Traditionally, a vincular pattern is written as a permutation with dashes inserted between terms that need not be adjacent and no dashes between terms that must be adjacent. Alternatively, and perhaps preferably, terms that must be adjacent are underlined. For example, contains two occurrences of (or ) and a single occurrence of (), but avoids ().
A vincular pattern in which all the terms must occur contiguously is known as a consecutive pattern.
In a bivincular pattern, conditions are also placed on which terms must take adjacent values.
Classical, vincular and bivincular patterns are all example of the more general family of mesh patterns. Formally, a mesh pattern of length is a pair with and a set of pairs of integers. The elements of identify the lower left corners of unit squares in the plot of , which specify forbidden regions. Mesh pattern is depicted by a figure consisting of the plot of with the forbidden regions shaded. See Figure 5 for an example.
An occurrence of mesh pattern in a permutation consists of an occurrence of the classical pattern in such that no elements of occur in the shaded regions of the figure. A vincular pattern is thus a mesh pattern in which complete columns shaded.
Sets of permutations defined by avoiding barred, vincular, bivincular or mesh patterns that are not closed under taking subpermutations are known as nonclassical permutation classes.
Growth rates
Given a permutation class , we use to denote the permutations of length in . It is natural to ask how quickly the sequence grows.
In proving the Stanley–Wilf Conjecture, Marcos and Tardos established that the growth of every classical permutation class except the class of all permutations is at most exponential. Hence, the upper growth rate and lower growth rate of a class are defined to be
The theorem of Marcos and Tardos states that and are both finite.
When , this quantity is called the proper growth rate (or just the growth rate) of and denoted . Principal classes, those of the form , are known to have proper growth rates. The growth rate of is sometimes known as the Stanley–Wilf limit of and denoted . It is widely believed, though not yet proven, that every classical permutation class has a proper growth rate.
Wilf equivalence
Given two classes, and , one natural question is to determine whether they are equinumerous, i.e. for every . Two permutation classes that are equinumerous are said to be Wilf equivalent and the equivalence classes are called Wilf classes. If principal classes and are Wilf equivalent, we simply say that and are Wilf equivalent.
From the graphical perspective, it is clear that classes related by symmetries of the square are Wilf equivalent. Thus, for example, , , and are equinumerous. However, not all Wilf equivalences are a result of these symmetries. Indeed, as is well known, both and are counted by the Catalan numbers, so all permutations of length three are in the same Wilf class.
Generating functions
The ordinary generating function of a permutation class is defined to be the formal power series
Thus, each permutation makes a contribution of , the result being that, for each , the coefficient of is the number of permutations of length . Clearly, two classes are Wilfequivalent if their generating functions are identical.
A generating function is rational if it is the ratio of two polynomials. A generating function is algebraic if it can be defined as the root of a polynomial equation. That is, there exists a bivariate polynomial such that .
References
 [1] Miklós Bóna. Combinatorics of Permutations. Discrete Mathematics and its Applications. CRC Press, second edition, 2012.
 [2] Sergey Kitaev. Patterns in Permutations and Words. Springer, 2011.