Growth rates of geometric grid classes of permutations

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 against those of the corresponding monotone grid classes, and present new results concerning the effect of edge subdivision on the largest root of the matching polynomial.


Introduction
Following the proof by Marcus and Tardos [12] of the Stanley-Wilf conjecture, there has been particular interest in the growth rates of permutation classes. Kaiser and Klazar [9] determined the possible growth rates less than 2, and then Vatter [17] characterised all the (countably many) permutation classes with growth rates below κ ≈ 2.20557 and established that there are uncountably many permutation classes with growth rate κ. Vatter also determined that there are permutation classes having every growth rate above λ ≈ 2.48188 [16]. (The behaviour between κ and λ is the subject of current research.) Critical to these results has been the consideration of grid classes of permutations, and particularly of geometric grid classes. Following initial work on particular geometric grid classes by Waton [20], Vatter and Waton [19], and Elizalde [4], their general structural properties have been investigated by Albert, Atkinson, Bouvel, Ruškuc and Vatter [1]. We build on their work to establish the growth rate of any given geometric grid class. Before we can state our result, we need a number of definitions (see [1] for a more detailed presentation).
A geometric grid class is specified by a 0/±1 matrix which represents the shape of plots of permutations in the class. To match the Cartesian coordinate system, we index these matrices from the lower left, by column and then by row. If M is such a matrix, then we say that the standard figure of M, denoted Λ M , is the subset of R 2 consisting of the union of oblique open line segments L i,j with slope M i,j for each i, j for which M i,j is nonzero, where L i,j extends from The geometric grid class Geom(M) is then defined to be the set of permutations σ 1 σ 2 . . . σ n that can be plotted as a subset of the standard figure, i.e. for which there exists a sequence of points (x 1 , y 1 ), . . . , (x n , y n ) ∈ Λ M such that x 1 < x 2 < . . . < x n and the sequence y 1 , . . . , y n is order-isomorphic to σ 1 , . . . , σ n . See Figure 1 for an example.
If g n is the number of permutations of length n in Geom(M), then the growth rate of the class is given by gr(Geom(M)) = lim n→∞ g 1/n n . We will demonstrate that this limit exists. 1 Much of the structure of a geometric grid class is reflected in a graph that we associate with the underlying matrix. If M is a 0/±1 matrix of dimensions t × u, the row-column graph G(M) of M is the bipartite graph with vertices r 1 , . . . , r t , c 1 , . . . , c u and an edge between r i and c j if and only if M i,j = 0. We label each edge r i c j with the value of M i,j . See Figure 1 for an example. We will demonstrate a connection between the growth rate of Geom(M) and the matching polynomial of the graph G(M ×2 ) -the row-column graph of the double refinement of M. A k-matching of a graph is a set of k edges, no pair of which have a vertex in common. For example, the negative (dashed) edges in the graph in Figure 2 constitute a 4-matching. If, for each k, m k (G) denotes the number of distinct k-matchings of a graph G with n vertices, then the matching polynomial µ G (z) of G is defined to be (1) An introduction to the matching polynomial can be found in Godsil and Gutman [6] and in the book by Lovász and Plummer [11].
With the relevant definitions complete, we can now state our theorem:

Griddings, consistent orientations and trace monoids
In order to prove our result, we relate geometric grid classes to trace monoids. This relationship was first used by Vatter and Waton [18] to establish certain structural properties of grid classes, and was developed further in [1] from where we use a number of results. To begin with, we need to consider griddings of permutations.
If M has dimensions t × u, then an M-gridding of a permutation σ 1 . . . σ n in Geom(M) consists of two sequences c 1 , . . . , c t and r 1 , . . . , r u such that there is some plot (x 1 , y 1 ), . . . , (x n , y n ) of σ for which c i is the number of points (x k , y k ) in column i (with i − 1 < x k < i), and r j is the number of points in row j (with j − 1 < y k < j). 2 Note that a permutation may have multiple distinct griddings in a given geometric grid class; see Figure 1 for an example. We call a permutation together with one of its M-griddings an M-gridded permutation. We use Geom # (M) to denote the set of all M-gridded permutations.
From an enumerative perspective, it can be much easier working with M-gridded permutations than directly with the permutations themselves. The following observation means that we can, in fact, restrict our considerations to M-gridded permutations: To determine the growth rate of Geom # (M), we will relate M-gridded permutations to words in a trace monoid. To achieve this, one additional concept is required, that of a consistent orientation of a standard figure. It is not always possible to consistently orient a standard figure. The ability to do so depends on the cycles in the row-column graph. We say that the parity of a cycle in G(M) is the product of the labels of its edges, a positive cycle is one which has parity 1, and a negative cycle is one with parity −1. The following result relates cycle parity to consistent orientations: For example, G 1 0 −1 1 −1 1 contains a negative cycle so its standard figure has no consistent orientation (see Figure 1), whereas G −1 0 −1 1 −1 1 has no negative cycles so its standard figure has a consistent orientation (see Figure 3). On the other hand, the row-column graph of the double refinement of a matrix never contains a negative cycle, so we have the following:  a 32 a 32 a 11 a 12 a 21 a 31 a 32 and a 11 a 32 a 21 a 32 a 31 a 12 a 32 . Both plots correspond to the same gridding.
We are now in a position to describe the association between words and M-gridded permutations. If M is a 0/±1 matrix, then we let Σ M = {a ij : M i,j = 0} be an alphabet of symbols -one for each nonzero cell in M. If we have a consistent orientation for Λ M , then we can associate to each finite word w 1 . . . w n over Σ M a specific plot of a permutation in Geom(M) as follows: If w k = a ij , include the point at distance k n+1 √ 2 along line segment L i,j according to its orientation. See Figure 3 for two examples. Clearly, this induces a mapping from the set of all finite words over Σ M to Geom # (M). In fact, it can readily be shown that this map is surjective, every M-gridded permutation corresponding to some word over Σ M ([1] Proposition 5.3).
As can be seen in Figure 3, distinct words may be mapped to the same gridded permutation. This occurs because the order in which two points are included is immaterial if they occur in cells that are neither in the same column nor in the same row. From the perspective of the words, adjacent symbols corresponding to such cells may be interchanged without changing the gridded permutation. This corresponds to a structure known as a trace monoid: If we have a consistent orientation for standard figure Λ M , then we define the trace monoid of M, which we denote by M(M), to be the set of equivalence classes of words over Σ M in which a ij and a kℓ commute (i.e. a ij a kℓ = a kℓ a ij ) whenever i = k and j = ℓ. It is then relatively straightforward to show equivalence between gridded permutations and elements of the trace monoid:

Proof of Theorem 1.2
Trace monoids were first studied by Cartier and Foata [3]. They determined the general form of the generating function, as follows: Since symbols in M(M) commute if and only if they correspond to cells that are neither in the same column nor in the same row, it is easy to see that r k (M) is the number of distinct ways of placing k chess rooks on the nonzero entries of M in such a way that no two rooks attack each other by being in the same column or row. The numbers r k (M) are known as the rook numbers for M (see Riordan [14]). Moreover, a matching in the row-column graph G(M) also corresponds to a set of cells no pair of which share a column or row. So the rook numbers for M are the same as the numbers of matchings in G(M): Now, by elementary analytic combinatorics, we know that the growth rate of M(M) is given by the reciprocal of the root of the denominator of f M (z) that has least magnitude (see [5] Theorem IV.7). The fact that this polynomial has a unique root of smallest modulus was proved by Goldwurm and Santini in [7]. It is real and positive by Pringsheim's Theorem.
But the reciprocal of the smallest root of a polynomial is the same as the largest root of the reciprocal polynomial (obtained by reversing the order of the coefficients). Hence, if M has dimensions t × u and n = t + u, then the growth rate of M(M) is the largest (positive real) root of the polynomial Here, g M (z) is the reciprocal polynomial of (f M (z)) −1 multiplied by some nonnegative power of z, since r k (M) = 0 for all k > ⌊n/2⌋. Note also that n is the number of vertices in G(M).
If we now compare the definition of g M (z) in (2) with that of the matching polynomial µ G (z) in (1) and use Observation 3.2, then we see that: Hence, the largest root of g M (z) is the square of the largest root of µ G(M) (z).
We now have all we need to prove Theorem 1.

Discussion
In this final section we briefly consider two issues and make some conjectures for future consideration. First, we investigate the effect of cycle parity on the growth rate. Then, we look at the relationship between the growth rate of a geometric grid class and the growth rate of the corresponding monotone grid class of permutations.
By Lemma 2.2, we know that if G(M) is free of negative cycles then Λ M can be consistently oriented. Thus, we have the following special case of Theorem 1.2: In general, the growth rate of a geometric grid class containing a negative cycle will be greater than the growth rate of the corresponding class without a negative cycle. For example, we have gr Geom 1 0 −1 whereas gr Geom −1 0 −1 In the specific case that G(M) is a cycle graph C n , we have gr(Geom(M)) =    4 cos 2 π 2n , if G(M) is a positive cycle; 4 cos 2 π 4n , if G(M) is a negative cycle.
If G(M) is a negative cycle, then G(M ×2 ) = C 2n . See Gutman [8] for the roots of µ C n (z).
We make the following conjecture on the existence of a strict inequality: We determined in [2] that the growth rate of monotone grid class Grid(M) equals the square of the spectral radius of G(M). The spectral radius of a graph is the largest root of its characteristic polynomial. Note that the growth rate of a monotone grid class is independent of the parity of its cycles. ). This is consistent with the fact that the characteristic polynomial of a graph is identical to its matching polynomial if and only if the graph is acyclic. (This result was discovered independently by Sachs [15], Mowshowitz [13] and Lovász and Pelikán [10].) In particular, if G is a forest, the roots of µ G (z) coincide with the spectrum of G.
Typically, the growth rate of a monotone grid class will be greater than the growth rate of the corresponding geometric grid class.  (3) and (4). We conclude by conjecturing that in the connected case we have a strict inequality: This would follow from the following (stronger) result:

Conjecture 4.4.
If a graph G is connected and contains a cycle, then the spectral radius of G strictly exceeds the largest root of the matching polynomial µ G (z).