Permutation Pattern Matching (PPM) is the problem of deciding for a given pair of permutations P and T whether the pattern P is contained in the text T. Bose, Buss and Lubiw showed that PPM is NP-complete. In view of this result, it is natural to ask how the situation changes when we restrict the pattern P to a fixed permutation class C; this is known as the C-Pattern PPM problem.
Grid classes are special kind of permutation classes, consisting of permutations admitting a grid-like decomposition into simpler building blocks. Of particular interest are the so-called monotone grid classes, in which each building block is a monotone sequence. Recently, it has been discovered that grid classes, especially the monotone ones, play a fundamental role in the understanding of the structure of general permutation classes. This motivates us to study the hardness of C-Pattern PPM for a (monotone) grid class C.
We provide a complexity dichotomy for C-Pattern PPM when C is taken to be a monotone grid class. Specifically, we show that the problem is polynomial-time solvable if a certain graph associated with C, called the cell graph, is a forest, and it is NP-complete otherwise. We further generalize our results to grid classes whose blocks belong to classes of bounded grid-width. We show that the C-Pattern PPM for such a grid class C is polynomial-time solvable if the cell graph of C avoids a cycle or a certain special type of path, and it is NP-complete otherwise.
We introduce and characterise grid classes, which are natural generalisations of other well-studied permutation classes. This characterisation allows us to give a new, short proof of the Fibonacci dichotomy: the number of permutations of length n in a permutation class is either at least as large as the nth Fibonacci number or is eventually polynomial.
We consider uniform random permutations in classes having a finite combinatorial specification for the substitution decomposition. These classes include (but are not limited to) all permutation classes with a finite number of simple permutations. Our goal is to study their limiting behavior in the sense of permutons. The limit depends on the structure of the specification restricted to families with the largest growth rate. When it is strongly connected, two cases occur. If the associated system of equations is linear, the limiting permuton is a deterministic $X$-shape. Otherwise, the limiting…
In this paper, we reveal an intriguing relationship between two seemingly unrelated notions: letter graphs and geometric grid classes of permutations. An important property common for both of them is well-quasi-orderability, implying, in a non-constructive way, a polynomial-time recognition of geometric grid classes of permutations and $k$-letter graphs for a fixed $k$. However, constructive algorithms are available only for $k=2$. In this paper, we present the first constructive polynomial-time algorithm for the recognition of $3$-letter graphs. It is based on a structural characterization of…
A grid class consists of permutations whose pictorial depiction can be partitioned into increasing and decreasing parts as determined by a given matrix. In this paper, we introduce a method for enumerating cyclic permutations in vector grid classes by establishing a bijective relationship with certain necklaces. We use this method to complete the enumeration of cyclic permutations in the length 3 vector grid classes. In addition, we define an analog of Wilf-equivalence between these sets. We conclude by discussing cyclic permutations in alternating grid classes.
A monotone grid class is a permutation class (i.e., a downset of permutations under the containment order) defined by local monotonicity conditions. We give a simplified proof of a result of Murphy and Vatter that monotone grid classes of forests are partially well-ordered.
We characterise those permutation classes whose simple permutations are monotone griddable. This characterisation is obtained by identifying a set of nine substructures, at least one of which must occur in any simple permutation containing a long sum of 21s.
This paper analyzes relations between pattern avoidance of certain permutations and graphs on staircase grids and boundary grids, and proves two conjectures posed by Bean, Tannock, and Ulfarsson (2015). More specifically, this paper enumerates a certain family of staircase encodings and proves that the downcore graph, a certain graph established on the boundary grid, is pure if and only if the permutation corresponding to the boundary grid avoids the classical patterns 123 and 2143.
We uncover a connection between two seemingly unrelated notions: lettericity, from structural graph theory, and geometric griddability, from the world of permutation patterns. Both of these notions capture important structural properties of their respective classes of objects. We prove that these notions are equivalent in the sense that a permutation class is geometrically griddable if and only if the corresponding class of inversion graphs has bounded lettericity.
Geometric grid classes of permutations have proven to be key in investigations of classical permutation pattern classes. By considering the representation of gridded permutations as words in a trace monoid, we prove that every geometric grid class has a growth rate which is given by the square of the largest root of the matching polynomial of a related graph. As a consequence, we characterise the set of growth rates of geometric grid classes in terms of the spectral radii of trees, explore the influence of "cycle parity" on the growth rate, compare the growth rates of geometric grid classes ag…
Bevan established that the growth rate of a monotone grid class of permutations is equal to the square of the spectral radius of a related bipartite graph. We give an elementary and self-contained proof of a generalization of this result using only Stirling's Formula, the method of Lagrange multipliers, and the singular value decomposition of matrices.
We study the complexity of the decision problem known as Permutation Pattern Matching, or PPM. The input of PPM consists of a pair of permutations $τ$ (the `text') and $π$ (the `pattern'), and the goal is to decide whether $τ$ contains $π$ as a subpermutation. On general inputs, PPM is known to be NP-complete by a result of Bose, Buss and Lubiw. In this paper, we focus on restricted instances of PPM where the text is assumed to avoid a fixed (small) pattern $σ$; this restriction is known as Av($σ$)-PPM. It has been previously shown that Av($σ$)-PPM is polynomial for any $σ$ of size at most 3, …
A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope \pm1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such sub…
Geometric grid classes and the substitution decomposition have both been shown to be fundamental in the understanding of the structure of permutation classes. In particular, these are the two main tools in the recent classification of permutation classes of growth rate less than $κ\approx2.20557$ (a specific algebraic integer at which infinite antichains begin to appear). Using language- and order-theoretic methods, we prove that the substitution closures of geometric grid classes are partially well-ordered, finitely based, and that all their subclasses have algebraic generating functions. We …
Characterizing sets of permutations whose associated quasisymmetric function is symmetric and Schur-positive is a long-standing problem in algebraic combinatorics. In this paper we present a general method to construct Schur-positive sets and multisets, based on geometric grid classes and the product operation. Our approach produces many new instances of Schur-positive sets, and provides a broad framework that explains the existence of known such sets that until now were sporadic cases.
A grid polygon is a polygon whose vertices are points of a grid. We define an injective map between permutations of length n and a subset of grid polygons on n vertices, which we call consecutive-minima polygons. By the kernel method, we enumerate sets of permutations whose consecutive-minima polygons satisfy specific geometric conditions. We deal with 2-variate and 3-variate generating functions involving derivatives, cases which are not routinely solved by the kernel method.
Monotone grid classes of permutations have proven very effective in helping to determine structural and enumerative properties of classical permutation pattern classes. Associated with grid class $\mathrm{Grid}(M)$ is a graph, $G(M)$, known as its "row-column" graph. We prove that the exponential growth rate of $\mathrm{Grid}(M)$ is equal to the square of the spectral radius of $G(M)$. Consequently, we utilize spectral graph theoretic results to characterise all slowly growing grid classes and to show that for every $γ\geqslant2+\sqrt{5}$ there is a grid class with growth rate arbitrarily clos…
We study aspects of the enumeration of permutation classes, sets of permutations closed downwards under the subpermutation order. First, we consider monotone grid classes of permutations. We present procedures for calculating the generating function of any class whose matrix has dimensions $m \times 1$ for some $m$, and of acyclic and unicyclic classes of gridded permutations. We show that almost all large permutations in a grid class have the same shape, and determine this limit shape. We prove that the growth rate of a grid class is given by the square of the spectral radius of an associated…
We enumerate the pattern class Av(2143,4231) and completely describe its permutations. The main tools are simple permutations and monotone grid classes.
A permutation of size $n$ can be identified to its diagram in which there is exactly one point per row and column in the grid $[n]^2$. In this paper we consider multidimensional permutations (or $d$-permutations), which are identified to their diagrams on the grid $[n]^d$ in which there is exactly one point per hyperplane $x_i=j$ for $i\in[d]$ and $j\in[n]$. We first investigate exhaustively all small pattern avoiding classes. We provide some bijection to enumerate some of these classes and we propose some conjectures for others. We then give a generalization of well-studied Baxter permutation…
We find the generating function for the class of all permutations that avoid the patterns 3124 and 4312 by showing that it is an inflation of the union of two geometric grid classes.
We prove necessary and sufficient conditions on a family of (generalised) gridding matrices to determine when the corresponding permutation classes are partially well-ordered. One direction requires an application of Higman's Theorem and relies on there being only finitely many simple permutations in the only non-monotone cell of each component of the matrix. The other direction is proved by a more general result that allows the construction of infinite antichains in any grid class of a matrix whose graph has a component containing two or more non-monotone-griddable cells. The construction use…
We enumerate three specific permutation classes defined by two forbidden patterns of length four. The techniques involve inflations of geometric grid classes.
We characterize all permutations on even-by-even grids that maximally separate neighboring vertices. More precisely, let $n_1$, $n_2$ be positive even integers, let $I(n_1,n_2)=\{1,\dots,n_1\}\times\{1,\dots,n_2\}$ be the $n_1\times n_2$ grid, let $d$ be the $L_1$ metric on $I(n_1,n_2)$, and let $N=\{\{x,y\}\in I(n_1,n_2)\times I(n_1,n_2):d(x,y)=1\}$ be the set of neighbors in $I(n_1,n_2)$. We characterize all permutations $π$ of $I(n_1,n_2)$ that maximize $\sum_{\{x,y\}\in N} d(π(x),π(y))$.
The packet routing problem plays an essential role in communication networks. It involves how to transfer data from some origins to some destinations within a reasonable amount of time. In the $(\ell,k)$-routing problem, each node can send at most $\ell$ packets and receive at most $k$ packets. Permutation routing is the particular case $\ell=k=1$. In the $r$-central routing problem, all nodes at distance at most $r$ from a fixed node $v$ want to send a packet to $v$. In this article we study the permutation routing, the $r$-central routing and the general $(\ell,k)$-routing problems on plane …
The simple permutations in two permutation classes --- the 321-avoiding permutations and the skew-merged permutations --- are enumerated using a uniform method. In both cases, these enumerations were known implicitly, by working backwards from the enumeration of the class, but the simple permutations had not been enumerated explicitly. In particular, the enumeration of the simple skew-merged permutations leads to the first truly structural enumeration of this class as a whole. The extension of this method to a wider collection of classes namely grid classes of infinite paths is discussed.
Permutation Pattern Matching (or PPM) is a decision problem whose input is a pair of permutations $π$ and $τ$, represented as sequences of integers, and the task is to determine whether $τ$ contains a subsequence order-isomorphic to $π$. Bose, Buss and Lubiw proved that PPM is NP-complete on general inputs. We show that PPM is NP-complete even when $π$ has no decreasing subsequence of length 3 and $τ$ has no decreasing subsequence of length 4. This provides the first known example of PPM being hard when one or both of $π$ and $σ$ are restricted to a proper hereditary class of permutations. Thi…
We establish that there is an algebraic number $ξ\approx 2.30522$ such that while there are uncountably many growth rates of permutation classes arbitrarily close to $ξ,$ there are only countably many less than $ξ$. Central to the proof are various structural notions regarding generalized grid classes and a new property of permutation classes called concentration. The classification of growth rates up to $ξ$ is completed in a subsequent paper.
We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number $κ$, approximately 2.20557, for which there are only countably many permutation classes of growth rate (Stanley-Wilf limit) less than $κ$ but uncountably many permutation classes of growth rate $κ$, answering a question of Klazar. We go on to completely characterize the possible sub-$κ$ growth rates of permutation classes, answering a question of Kaiser and Klazar. Central to our proofs are the concepts of generalized grid classes (introduced …
We introduce an algorithm that conjectures the structure of a permutation class in the form of a disjoint cover of "rules"; similar to generalized grid classes. The cover is usually easily verified by a human and translated into an enumeration. The algorithm is successful on different inputs than other algorithms and can succeed with any polynomial permutation class. We apply it to every non-polynomial permutation class avoiding a set of length four patterns. The structures found by the algorithm can sometimes allow an enumeration of the permutation class with respect to permutation statistics…
We study subclasses of grid intersection graphs from the perspective of order dimension. We show that partial orders of height two whose comparability graph is a grid intersection graph have order dimension at most four. Starting from this observation we provide a comprehensive study of classes of graphs between grid intersection graphs and bipartite permutation graphs and the containment relation on these classes. Order dimension plays a role in many arguments.
In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain four classical Gray codes for permutations, bitstrings, binary trees and set partitions as special cases. We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different fami…
We show that, given a suitable combinatorial specification for a permutation class $\mathcal{C}$, one can obtain a specification for the juxtaposition (on either side) of $\mathcal{C}$ with Av(21) or Av(12), and that if the enumeration for $\mathcal{C}$ is given by a rational or algebraic generating function, so is the enumeration for the juxtaposition. Furthermore this process can be iterated, thereby providing an effective method to enumerate any 'skinny' $k\times 1$ grid class in which at most one cell is non-monotone, with a guarantee on the nature of the enumeration given the nature of th…
By extending the notion of grid classes to include infinite grids, we establish a structural characterisation of the simple permutations in Av(4231, 35142, 42513, 351624), a pattern class which has three different connections with algebraic geometry, including the specification of indices of Schubert varieties defined by inclusions. This characterisation leads to the enumeration of the class.
A permutation is centrosymmetric if it is fixed by a half-turn rotation of its diagram. Initially motivated by a question by Alexander Woo, we investigate the question of whether the growth rate of a permutation class equals the growth rate of its even-size centrosymmetric elements. We present various examples where the latter growth rate is strictly less, but we conjecture that the reverse inequality cannot occur. We conjecture that equality holds if the class is sum closed, and we prove this conjecture in the special case where the growth rate is at most $ξ\approx 2.30522$, using results fro…
We show that the Graph Isomorphism (GI) problem and the related problems of String Isomorphism (under group action) (SI) and Coset Intersection (CI) can be solved in quasipolynomial ($\exp((\log n)^{O(1)})$) time. The best previous bound for GI was $\exp(O(\sqrt{n\log n}))$, where $n$ is the number of vertices (Luks, 1983); for the other two problems, the bound was similar, $\exp(\tilde{O}(\sqrt{n}))$, where $n$ is the size of the permutation domain (Babai, 1983). The algorithm builds on Luks's SI framework and attacks the barrier configurations for Luks's algorithm by group theoretic "local c…
The structure of three pattern classes Av(2143, 4321), Av(2143, 4312) and Av(1324, 4312) is determined using the machinery of monotone grid classes. This allows the permutations in these classes to be described in terms of simple diagrams and regular languages and, using this, the rational generating functions which enumerate these classes are determined.
Correspondence homomorphisms are both a generalization of standard homomorphisms and a generalization of correspondence colourings. For a fixed target graph $H$, the problem is to decide whether an input graph $G$, with each edge labeled by a pair of permutations of $V(H)$, admits a homomorphism to $H$ `corresponding' to the labels, in a sense explained below. We classify the complexity of this problem as a function of the fixed graph $H$. It turns out that there is dichotomy -- each of the problems is polynomial-time solvable or NP-complete. While most graphs $H$ yield NP-complete problems, t…
We establish an improved lower bound of 10.271 for the exponential growth rate of the class of permutations avoiding the pattern 1324, and an improved upper bound of 13.5. These results depend on a new exact structural characterisation of 1324-avoiders as a subclass of an infinite staircase grid class, together with precise asymptotics of a small domino subclass whose enumeration we relate to West-two-stack-sortable permutations and planar maps. The bounds are established by carefully combining copies of the dominoes in particular ways consistent with the structural characterisation. The lower…
