Permutation pattern

In combinatorial mathematics and theoretical computer science, a permutation pattern is a sub-permutation of a longer permutation. A permutation π of length n is written as a word in one-line notation (i.e., in two-line notation with the first line omitted) as π = π₁π₂…π_n, where π_i is the i^th number in the word. For example, in the permutation π = 391867452, π₁=3 and π₉=2. A permutation π is said to contain the permutation σ if there exists a subsequence of (not necessarily consecutive) entries of π that has the same relative order as σ, and in this case σ is said to be a pattern of π, written σ ≤ π. Otherwise, π is said to avoid the permutation σ. For example, the permutation π = 391867452 contains the pattern σ = 51342, as can be seen in the highlighted subsequence of π = 391867452 (or π = 391867452 or π = 391867452 or π = 391867452). Each subsequence (91674, 91675, 91672, 91452) is called a copy, instance, or occurrence of σ. Since the permutation π = 391867452 contains no increasing subsequence of length four, π avoids 1234.

Early results

A case can be made that Percy MacMahon (1915) was the first to prove a result in the field with his study of "lattice permutations". In particular MacMahon shows that the permutations which can be divided into two decreasing subsequences (i.e., the 123-avoiding permutations) are counted by the Catalan numbers.

Another early landmark result in the field is the Erdős–Szekeres theorem; in permutation pattern language, the theorem states that for any positive integers a and b every permutation of length at least $(a-1)(b-1) + 1$ must contain either the pattern $1, 2, 3,\dots , a$ or the pattern $b,b-1,\dots , 2, 1$.

Computer science origins

The study of permutation patterns began in earnest with Donald Knuth's consideration of stack-sorting in 1968. Knuth showed that the permutation π can be sorted by a stack if and only if π avoids 231, and that the stack-sortable permutations are enumerated by the Catalan numbers. Knuth also raised questions about sorting with deques. In particular, Knuth's question asking how many permutation of n elements are obtainable with the use of a deque remains open. Shortly thereafter, Robert Tarjan (1972) investigated sorting by networks of stacks, while Vaughan Pratt (1973) showed that the permutation π can be sorted by a deque if and only if for all k, π avoids 5,2,7,4,...,4k+1,4k−2,3,4k,1, and 5,2,7,4,...,4k+3,4k,1,4k+2,3, and every permutation that can be obtained from either of these by interchanging the last two elements or the 1 and the 2. Because this collection of permutations is infinite (in fact, it is the first published example of an infinite antichain of permutations), it is not immediately clear how long it takes to decide if a permutation can be sorted by a deque. Rosenstiehl & Tarjan (1984) later presented a linear (in the length of π) time algorithm which determines if π can be sorted by a deque.

In his paper, Pratt remarked that this permutation pattern order “seems to be the only partial order on permutation that arises in a simple and natural way” and concludes by noting that “from an abstract point of view”, the permutation pattern order “is even more interesting than the networks we were characterizing”.

Enumerative origins

Main article: Enumerations of specific permutation classes

Many problems in the study of permutation patterns are pursued using the tools of the field of enumerative combinatorics. Two central problems are Wilf-equivalence and the Stanley-Wilf conjecture.

Wilf-equivalence

One central problem in pattern avoidance that has been studied in the last 40 years is that of determining the number of permutations of length n which avoid a fixed permutation β. Let Av_n(β) denote the set of permutations of length n which avoid β, and let |Av_n(β)| denote the number of such permutations. Two patterns β and σ are said to be Wilf-equivalent if |Av_n(β)| = |Av_n(σ)| for all n.

A basic example is if π = π₁π₂…π_n avoids 321, i.e., has no decreasing subsequence of length 3, then its reverse, π' = π_nπ_n-1…π₁ avoids 123, i.e., has no increasing subsequence of length 3. It follows that, for each n, there are the same number of patterns of length n which avoid the patterns 123 and 321, i.e., 123 and 321 are Wilf-equivalent. By symmetry we see that the patterns 132, 231, 213, 312 are Wilf-equivalent: If π = π₁π₂…π_n avoids 132, then its reverse, π' = π_nπ_n-1…π₁ avoids 231, its complement π" = (n+1-π₁)(n+1-π₂)…(n+1-π_n) avoids 312, and the reverse of π" avoids 213. Since these operations are bijective, there are the same number of permutations of length n which avoid 132, 231, 213, and 312.

One way in which symmetry classes of permutations are commonly visualized is by using permutation matrices. Let M(π) be the permutation matrix having a 1 in position (i,π(i)) for 1 ≤ i ≤ n and 0's elsewhere. Then a permutation p contains the pattern π precisely when the matrix M(p) contains M(π) as a submatrix. The bijective reverse operation just described corresponds to a flip of the permutation matrix across a vertical axis; the bijective complement operation above corresponds to a flip of the permutation matrix across a horizontal axis. There is also a bijective operation which takes a permutation π = π₁π₂…π_n to another permutation via the map which takes (i,π_i) to (π_i, i), which corresponds to a reflection across a diagonal of the permutation matrix. The symmetry class of p is the set of all permutations of p obtained by a finite number of reflections across vertical, horizontal, and diagonal axes. In other words, the symmetry class of p is the orbit of an action by the Dihedral group D₈ on M(p). Within each symmetry class, permutations are Wilf-equivalent.

There are also numerous examples of nontrivial Wilf-equivalences, including |Av_n(123)| = |Av_n(231)|. This was proved by combining the works of MacMahon and Knuth, who showed that for a given n these sets each have size C_n, the n^th Catalan number. Simion & Schmidt (1985) gave the first bijective proof that 123- and 231-avoiding permutations are equinumerous. It follows that all permutations of length 3 are Wilf-equivalent, meaning there is one Wilf-equivalence class for permutations of length 3. Many other bijections between sets of permutations which avoid a pattern in the symmetry class {123,321} and sets of permutations which avoid a pattern in the symmetry class {132,231,213,312} have been given; see Claesson & Kitaev (2008) or Kitaev (2011) for a survey.

Classifying patterns of length 4 required novel approaches leading to other nontrivial Wilf-equivalences:

• Stankova (1994) proved that the permutations 1342 and 2413 are Wilf-equivalent.
• Stankova & West (2002) proved that for any permutation β, the permutations 231 ⊕ β and 312 ⊕ β are Wilf-equivalent, where ⊕ denotes the direct sum operation.
• Backelin, West & Xin (2007) proved that for any permutation β and any positive integer m, the permutations 12..m ⊕ β and m...21 ⊕ β are Wilf-equivalent.

There are 7 symmetry classes of permutations of length 4: {1234,4321}, {1324,4231}, {2143,3412}, {2413,3142}, {1243,2134,3421,4312}, {1432,2341,3214,4123}, and {1342,1423,2314,2431,3124,3241,4132,4213}. These symmetry classes together with the aforementioned Wilf-equivalences determine three Wilf-equivalence classes for patterns of length 4, namely {1234,1243,1432,2134,2143,2341,3142,3214,3412,4123,4312,4321}, {1342,1423,2314,2413,3142,2431,3124,3241,4132,4213}, and {1324,4231}. The following are the 3 different sequences |Av_n(β)| where β is of length four:

β	sequence enumerating Avn(β)	OEIS reference	exact enumeration reference
1342	1, 2, 6, 23, 103, 512, 2740, 15485, 91245, 555662, ...	A022558	Bóna (1997)
1234	1, 2, 6, 23, 103, 513, 2761, 15767, 94359, 586590, ...	A005802	Gessel (1990)
1324	1, 2, 6, 23, 103, 513, 2762, 15793, 94776, 591950, ...	A061552	unenumerated

Stanley-Wilf conjecture

In the late 1980s, Richard P. Stanley and Herbert Wilf conjectured that for every permutation β, there is some constant K such that |Av_n(β)| < Kⁿ. This was known as the Stanley–Wilf conjecture until it was proved by Adam Marcus and Gabor Tardos.

Poset and its Möbius Function

The set of all permutations of n with the definition that π ≤ p if π occurs as a pattern in p is a poset. If π occurs as a pattern in p but π ≠ p, then we write π < p. A closed interval of a poset P is defined whenever s ≤ t as $[s,t]=\{u\in P:s \leq u \leq t\}$ and $[s,t]=\emptyset$ if $s\nleq t$ in P.

The Möbius function (see incidence algebra) is an important poset invariant (mathematics), and Wilf first expressed the problem of computing the Möbius function of the permutation pattern poset in Wilf (2002). For a given interval [s,t] of a poset P, its Möbius function is defined recursively:

$\mu(x,y) = \begin{cases} {}\qquad 1 & \textrm{if}\quad x = y\\[6pt] \displaystyle -\sum_{z : x\leq z <y} \mu(x,z) & \textrm{for} \quad x<y \\[6pt] {}\qquad 0 & \textrm{otherwise}. \end{cases}$

Calculating values of the Möbius function for permutations within an interval [s,t] of a poset P becomes difficult as the lengths of s and t increase. Burstein et al. (2011) give a computationally efficient formula for the Möbius function of an interval [s,t] for which s and t are both separable permutations. Two consequences of this formula derived in Burstein et al. (2011) are that if s is separable, then μ(1,s)∈{0,1,-1}; and if s and t are separable, then |μ(s,t)| is at most the number of occurrences of s in t.

Closed classes

A closed class, also known as a pattern class, permutation class, or simply class of permutations is a downset in the permutation pattern order. Every class can be defined by the minimal permutations which do not lie inside it, its basis. Thus the basis for the stack-sortable permutations is {231}, while the basis for the deque-sortable permutations is infinite. The generating function for a class is Σ x^|π| where the sum is taken over all permutations π in the class.

Given a class of permutations, there are numerous questions that one may seek to answer, such as:

• What is the enumeration of the class?

• Does the class have a rational/algebraic/holonomic generating function?

• What is the growth rate of the class? (Or, if this does not exist, the upper or lower growth rate.)

• Is the basis of the class finite or infinite?

• Is the class partially well-ordered?

• Does the class satisfy the joint embedding property? (Classes which satisfy this are often called atomic.)

• How quickly can the membership problem for this class be decided? I.e., given a permutation π of length n, how long does it take to determine if π lies in the class?

General techniques to answer these questions are few and far between.

Packing densities

The permutation π is said to be β-optimal if no permutation of the same length as π has more copies of β. In his address to the SIAM meeting on Discrete Mathematics in 1992, Wilf defined the packing density of the permutation β of length k as

$\lim_{n\rightarrow\infty} \frac{\text{number of copies of }\beta\text{ in a }\beta\text{-optimal permutation of length }n}{\displaystyle{n\choose k}}.$

An unpublished argument of Fred Galvin shows that the quantity inside this limit is nonincreasing for n ≥ k, and so the limit exists. When β is monotone, its packing density is clearly 1, and packing densities are invariant under the group of symmetries generated by inverse and reverse, so for permutations of length three, there is only one nontrivial packing density. Walter Stromquist (unpublished) settled this case by showing that the packing density of 132 is 2√3 − 3, approximately 0.46410.

For permutations β of length four, there are (due to symmetries) seven cases to consider:

β	packing density	reference
1234	1	trivial
1432	root of x3 − 12x2 + 156x − 64 ≅ 0.42357	Price (1997)
2143	⅜ = 0.375	Price (1997)
1243	⅜ = 0.375	Albert et al. (2002)
1324	conjectured to be ≅ 0.244
1342	conjectured to be ≅ 0.19658
2413	conjectured to be ≅ 0.10474

For the three unknown permutations, there are bounds and conjectures. Price (1997) used an approximation algorithm which suggests that the packing density of 1324 is around 0.244. Birzhan Batkeyev (unpublished) constructed a family of permutations showing that the packing density of 1342 is at least the product of the packing densities of 132 and 1432, approximately 0.19658. This is conjectured to be the precise packing density of 1342. Presutti & Stromquist (2010) provided a lower bound on the packing density of 2413. This lower bound, which can be expressed in terms of an integral, is approximately 0.10474, and conjectured to be the true packing density.

Generalizations

There are several ways in which this notion of permutation patterns may be generalized. For example, a vincular pattern is a permutation containing dashes indicating the entries that need not occur consecutively (in the normal pattern definition, no entries need to occur consecutively). For example, the permutation 314265 has two copies of the dashed pattern 2-31-4, given by the entries 3426 and 3425. For a dashed pattern β and any permutation π, we write β(π) for the number of copies of β in π. Thus the number of inversions in π is 2-1(π), while the number of descents is 21(π). Going further, the number of valleys in π is 213(π) + 312(π), while the number of peaks is 231(π) + 132(π). These patterns were introduced by Babson & Steingrímsson (2000), who showed that almost all known Mahonian statistics could be expressed in terms of vincular permutations. For example, the Major index of π is equal to 1-32(π) + 2-31(π) + 3-21(π) + 21(π).

Another generalization is that of a barred pattern, in which some of the entries are barred. For π to avoid the barred pattern β means that every set of entries of π which form a copy of the nonbarred entries of β can be extended to form a copy of all entries of β. West (1993) introduced these types of patterns in his study of permutations which could be sorted by passing them twice through a stack. (Note that West's definition of sorting twice through a stack is not the same as sorting with two stacks in series.) Another example of barred patterns occurs in the work of Bousquet-Mélou & Butler (2007), who showed that the Schubert variety corresponding to π is locally factorial if and only if π avoids 1324 and 21354.

Superpatterns

Main article: Superpattern

A k-superpattern is a permutation of minimal length that contains all permutation patterns of length k. For example, consider the permutation 25314, which has length 5. It has 10 subsequences of length 3, including 253, 251, 234, 531, 534 and 314. (That is, we take 3 entries at a time, keeping the order preserved.) In each subsequence, we replace the smallest entry with 1, the next largest entry with 2, and the largest with 3. Thus, 253 gets renumbered to 132, 251 to 231, and so on. Applying this operation to the six selected subsequences gives all six possible length-3 patterns, 132, 231, 123, 321, 312 and 213. Thus, 25314 is a 3-superpattern.

It is known that a superpattern must have length at least k²/e², where e ≈ 2.71828 is Euler's number, and that there exist superpatterns of length k(k + 1)/2. The k(k + 1)/2 bound is conjectured to be within lower-order terms of the best possible value.

Computational complexity

Given a permutation $\tau$ (called the text) of length $n$ and another permutation $\pi$ (called the pattern), the permutation pattern matching (PPM) problem asks whether $\pi$ is contained in $\tau$. When both $n$ and $k$ are regarded as variables, the problem is known to be NP-complete, and the problem of counting the number of such matches is #P-complete. However, PPM can be solved in linear time when k is a constant. Indeed, Guillemot and Marx showed that PPM can be solved in time $2^{O(k^2\log k)} \cdot n$, meaning that it is fixed-parameter tractable with respect to $k$.

There are several variants on the PPM problem, as surveyed by Bruner and Lackner. For example, if the match is required to consist of contiguous entries then the problem can be solved in polynomial time.

Another variant is when both the pattern and text are restricted to a proper permutation class $\mathcal{C}$, in which case the problem is called $\mathcal{C}$-PPM. For example, Guillemot and Vialette showed that $\mbox{Av}(321)$-PPM could be solved in $O(k^2n^6)$ time. Albert, Lackner, Lackner, and Vatter later lowered this to $O(kn)$ and showed that the same bound holds for the class of skew-merged permutations. They further asked if the $\mathcal{C}$-PPM problem can be solved in polynomial time for every fixed proper permutation class $\mathcal{C}$.

See also

• Superpermutation

External links

A conference on permutation patterns has been held annually since 2003:

• Permutation Patterns 2003, February 10–14, 2003, University of Otago, Dunedin, New Zealand.
• Permutation Patterns 2004, July 5–9, 2004, Malaspina University-College, Nanaimo, British Columbia, Canada.
• Permutation Patterns 2005, March 7–11, 2005, University of Florida, Gainesville, Florida, USA.
• Permutation Patterns 2006, June 12–16, 2006, Reykjavík University, Reykjavík, Iceland.
• Permutation Patterns 2007, June 11–15, 2007, University of St. Andrews, St. Andrews, Scotland.
• Permutation Patterns 2008, June 16–20, 2008, University of Otago, Dunedin, New Zealand.
• Permutation Patterns 2009, July 13–17, 2009, Università di Firenze, Florence, Italy.
• Permutation Patterns 2010, August 9–13, 2010, Dartmouth College, Hanover, New Hampshire, USA.
• Permutation Patterns 2011, June 20–24, 2011, California Polytechnic State University, San Luis Obispo, California, USA.
• Permutation Patterns 2012, June 11–15, 2012, University of Strathclyde, Glasgow, Scotland.
• Permutation Patterns 2013, July 1–5, 2013, Université Paris Diderot, Paris, France.
• Permutation Patterns 2014, July 7–11, 2014, East Tennessee State University, Johnson City, Tennessee, USA.
• Permutation Patterns 2015, June 15–19, 2015, De Morgan House, London, England.
• Permutation Patterns 2016, June 27 – July 1, 2016, Howard University, Washington, DC, USA.

A Special Session on Patterns in Permutations and Words was held at the Spring Eastern Sectional meeting of the American Mathematical Society, March 7–8, 2015, Georgetown University, Washington, DC, USA.

Other links:

• Database of Permutation Pattern Avoidance, maintained by Bridget Tenner.