Packing Densities of Colored and Non-Colored Patterns

Pattern packing concerns finding an optimal permutation that contains the maximum number of occurrences of a given pattern and computing the corresponding packing density. In many instances such an optimal permutation can be characterized directly and the number of occurrences of the pattern in interest may be enumerated explicitly. In more complicated patterns a direct characterization may be more challenging, however computational results for long permutations can help provide an indirect characterization of the general form of an optimal permutation. Much work has been done on the study of pattern packing in layered patterns, as the optimal permutation of a layered pattern is easily characterized. It has been shown that there always exists a layered optimal permutation of a given layered pattern. Because all length three-patterns and all but two (under equivalence) length-four patterns are layered, this result solves the pattern packing problem for many simple patterns. A broader class of permutations called colored permutations is formed by assigning permuted elements a color from a corresponding color set. We explore the consequences of colored permutations and patterns on the pattern packing problem. Through examining the novel concept of colored blocks within a colored pattern or permutation, we present analogous results on optimal colored permutations of patterns containing two or three colored blocks. We also conjecture an extended result for patterns containing more than three colored blocks. The results we present encompass a broader class of patterns than the analogous layered patterns, with limitations first arising when a colored block contains a consecutively monochromatic non-layered pattern. From numerical observations that colored patterns are refinements of their associated non-colored patterns, we also present an explicit relationship between packing densities of colored patterns and their consecutively monochromatic constituents. This result is also conjectured to hold for any colored pattern containing any number of colored blocks

On Packing Densities of Set Partitions

We study packing densities for set partitions, which is a generalization of packing words. We use results from the literature about packing densities for permutations and words to provide packing densities for set partitions. These results give us most of the packing densities for partitions of the set $\{1,2,3\}$. In the final section we determine the packing density of the set partition $\{\{1,3\},\{2\}\}$.Comment: 12 pages, to appear in the Permutation Patterns edition of the Australasian Journal of Combinatoric

Waiting Time Distribution for the Emergence of Superpatterns

Consider a sequence X_1, X_2,... of i.i.d. uniform random variables taking values in the alphabet set {1,2,...,d}. A k-superpattern is a realization of X_1,...,X_t that contains, as an embedded subsequence, each of the non-order-isomorphic subpatterns of length k. We focus on the non-trivial case of d=k=3 and study the waiting time distribution of tau=inf{t>=7: X_1,...,X_t is a superpattern}Comment: 17 page

Packing sets of patterns

AbstractPacking density is a permutation occurrence statistic which describes the maximal number of permutations of a given type that can occur in another permutation. In this article we focus on containment of sets of permutations. Although this question has been tangentially considered previously, this is the first article focusing exclusively on it. We find the packing density for various special sets of permutations and study permutation and pattern co-occurrence

The feasible regions for consecutive patterns of pattern-avoiding permutations

We study the feasible region for consecutive patterns of pattern-avoiding permutations. More precisely, given a family $\mathcal C$ of permutations avoiding a fixed set of patterns, we study the limit of proportions of consecutive patterns on large permutations of $\mathcal C$. These limits form a region, which we call the \emph{pattern-avoiding feasible region for $\mathcal C$}. We show that, when $\mathcal C$ is the family of $\tau$-avoiding permutations, with either $\tau$ of size three or $\tau$ a monotone pattern, the pattern-avoiding feasible region for $\mathcal C$ is a polytope. We also determine its dimension using a new tool for the monotone pattern case, whereby we are able to compute the dimension of the image of a polytope after a projection. We further show some general results for the pattern-avoiding feasible region for any family $\mathcal C$ of permutations avoiding a fixed set of patterns, and we conjecture a general formula for its dimension. Along the way, we discuss connections of this work with the problem of packing patterns in pattern-avoiding permutations and to the study of local limits for pattern-avoiding permutations.Comment: New version before submission to journa

Combinatorial Optimization of Subsequence Patterns in Words

Packing patterns in words concerns finding a word with the maximum number of a prescribed pattern. The majority of the work done thus far is on packing patterns into permutations. In 2002, Albert, Atkinson, Handley, Holton and Stromquist showed that there always exists a layered permutation containing the maximum number of a layered pattern among all permutations of length n. Consequently, the packing density for all but two (up to equivalence) permutation patterns up to length 4 can be obtained. In this thesis we consider the analogous question for colored patterns and permutations. By introducing the concept of colored blocks we characterize the optimal permutations with the maximum number of a given colored pattern when it contains at most three colored blocks. As examples, we apply this characterization to find the optimal permutations of various colored patterns and subsequently obtain their corresponding packing densities

From Permutation Patterns to the Periodic Table

(The above abstract has been extracted by the translator from the original article (L. Pudwell, From Permutation Patterns to the Periodic Table, Notices of the American Mathematical Society, 67 994–1001.))Abstract: Permutation patterns is a burgeoning area of research with roots in enumerative combinatorics and theoretical computer science. This article first presents a brief overview of pattern avoidance and a survey of enumeration results that are standard knowledge within the field. Then, we turn our attention to a newer optimization problem of pattern packing. We survey pattern packing results in the general case before we consider packing in a specific type of permutation that leads to a new and surprising connection with physical chemistry. Note that the original paper has published in ``Notices of the American Mathematical Society, 67, Number 7, 994-1001" and we have translated it into Farsi. This is just an extended abstract for Journal of Mathematics and Society.  1. IntroductionLet $S_k$ be the set of all permutations on $[k]=\{1, 2,\ldots,k\}$. Given $\pi \in S_k$ and $\rho \in S_l$, we say that $\pi$ contains $\rho$ as a pattern if there exist $1\leq i_1\pi_310?$ What other chemical or physical structures can be described in terms of pattern packing or pattern avoidance? Are there other combinatorial structures that give alternate ways to generate the sequences of atomic numbers of particular groups of chemical elements? The variety of applications of permutation patterns has grown tremendously in recent decades, and modeling electron orbitals can now be added to the list

Density maximizers of layered permutations

A permutation is layered if it contains neither 231 nor 312 as a pattern. It is known that, if σ is a layered permutation, then the density of σ in a permutation of order n is maximized by a layered permutation. Albert, Atkinson, Handley, Holton and Stromquist [Electron. J. Combin. 9 (2002), #R5] claimed that the density of a layered permutation with layers of sizes (a,1,b) where a,b≥2 is asymptotically maximized by layered permutations with a bounded number of layers, and conjectured that the same holds if a layered permutation has no consecutive layers of size one and its first and last layers are of size at least two. We show that, if σ is a layered permutation whose first layer is sufficiently large and second layer is of size one, then the number of layers tends to infinity in every sequence of layered permutations asymptotically maximizing the density of σ. This disproves the conjecture and the claim of Albert et al. We complement this result by giving sufficient conditions on a layered permutation to have asymptotic or exact maximizers with a bounded number of layers