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A permutation of n letters is k-prolific if each (n − k)-subset of the letters in its one-line notation forms a unique pattern. We present a complete characterization of kprolific permutations for each k, proving that k-prolific permutations of m letters exist for every m k 2 /2 + 2k + 1, and that none exist of smaller size. Key to these results is a natural bijection between k-prolific permutations and certain "permuted" packings of diamonds.

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 Grid(M ) is a graph, G(M ), known as its "row-column" graph. We prove that the exponential growth rate of 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 2 + √ 5 there is a grid class with growth rate arbitrarily close to γ. To prove our main result, we establish bounds on the size of certain families of tours on graphs. In the process, we prove that the family of tours of even length on a connected graph grows at the same rate as the family of "balanced" tours on the graph (in which the number of times an edge is traversed in one direction is the same as the number of times it is traversed in the other direction). 5863 Licensed to Univ of Mississippi. Prepared on Tue Jul 14 02:56:52 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5864 DAVID BEVANClearly, the containment relation is a partial order on the set of all permutations. A classical permutation class (or "pattern class") is a set of permutations closed downwards (a down-set) in this partial order. From a graphical perspective, this means that erasing points from the plot of a permutation in a permutation class C always results in the plot of another permutation in C when the axes are rescaled appropriately.Given a permutation class C, we denote by C k = {σ ∈ C : |σ| = k} the set of permutations in C of length k. The (ordinary) generating function of C is thus k∈N |C k |z k = σ∈C z |σ| . It is common to define a permutation class C "negatively" by stating the minimal set of permutations B that do not occur in the class. In this case, we write C = Av(B) (where Av signifies "avoids"). B is called the basis of C. The basis of a permutation class is an antichain (a set of pairwise incomparable elements) and may be infinite. Licensed to Univ of Mississippi. Prepared on Tue Jul 14 02:56:52 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GROWTH RATES OF PERMUTATION GRID CLASSES 5865 basis has finite lower and upper exponential growth rates defined, respectively, by gr(C) = lim inf k→∞ |C k | 1/k and gr(C) = lim sup k→∞ |C k | 1/k .If the lower and upper growth rates coincide, then C has a growth rate, which we denote gr(C). (It is widely conjectured that every permutation class has a growth rate.) In [30], Vatter investigated the possible values of permutation class growth rates, and used generalised grid classes to characterize all the (countably many) permutation classes with growth rates below κ ≈ 2.20557. He also established that there are uncountably many permutation classes with growth rate κ, and in a separate...

The class Av(1324), of permutations avoiding the pattern 1324, is one of the simplest sets of combinatorial objects to define that has, thus far, failed to reveal its enumerative secrets. By considering certain large subsets of the class, which consist of permutations with a particularly regular structure, we prove that the growth rate of the class exceeds 9.81. This improves on a previous lower bound of 9.47. Central to our proof is an examination of the asymptotic distributions of certain substructures in the Hasse graphs of the permutations. In this context, we consider occurrences of patterns in Łukasiewicz paths and prove that in the limit they exhibit a concentrated Gaussian distribution.

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.