Set Of Permutations Research Articles

Fragmentary Structures in a Two-Dimensional Strip Packing Problem

The general problem of two-dimensional packing in a semi-bounded strip is considered. It is shown that the problem can be considered as an optimization problem on a fragmentary structure and is reduced to the problem of combinatorial optimization on a set of permutations. A universal approach to representing two-dimensional figures and an algorithm for packing them in a strip are considered. An approach to modifying the original problem to attain an optimal solution is proposed.

Permutation patterns in genome rearrangement problems: The reversal model

In the context of the genome rearrangement problem, we analyze two well known models, namely the reversal and the prefix reversal models, by exploiting the connection with the notion of permutation patterns. More specifically, for any k, we provide a characterization of the set of permutations having distance less than or equal to k from the identity (which is known to be a permutation class) in terms of what we call generating peg permutations and we describe some properties of its basis, which allow us to compute such a basis for small values of k.

A new heuristics for the event ordering in binary decision diagram applied in fault tree analysis

Fault tree is a common approach in probabilistic risk assessment of complex engineering systems. Since their introduction, binary decision diagrams proved to be a valuable tool for complete quantification of hard fault tree models. As is known, the size of the binary decision diagram representation is mainly determined by the quality of the selected fault tree event ordering scheme. Finding the optimal event ordering for binary decision diagram representation is a computationally intractable problem, for which reason heuristic approaches are applied to find reasonable good ordering schemes. The existing method for finding optimal ordering schemes related to special types of fan-in 2 read-once formulas is employed in our research to develop a new heuristic for fault tree. Various fault tree simplification methods are used for the sake of reducing fault tree model discrepancy from fan-in 2 read-once formulas. The reduced fault tree is traversed in a depth-first manner, as for every gate, the best ordering scheme is chosen from selected sets of input permutations. The quality of the final event ordering scheme is compared to orderings produced with depth-first left most heuristic on a set of fault tree models addressed in the literature as well as on a set of our hard models. Our method proves to be a useful heuristic for finding good static event ordering, and it compares favourably to the known heuristic based on a depth-first left most assignment approach.

Constructing permutation arrays using partition and extension

We give new lower bounds for M(n, d), for various positive integers n and d with $$n>d$$, where M(n, d) is the largest number of permutations on n symbols with pairwise Hamming distance at least d. Large sets of permutations on n symbols with pairwise Hamming distance d are needed for constructing error correcting permutation codes, which have been proposed for power-line communications. Our technique, partition and extension, is universally applicable to constructing such sets for all n and all d, $$d<n$$. We describe three new techniques, sequential partition and extension, parallel partition and extension, and a modified Kronecker product operation, which extend the applicability of partition and extension in different ways. We describe how partition and extension gives improved lower bounds for $$M(n,n-1)$$ using mutually orthogonal Latin squares (MOLS). We present efficient algorithms for computing new partitions: an iterative greedy algorithm and an algorithm based on integer linear programming. These algorithms yield partitions of positions (or symbols) used as input to our partition and extension techniques. We report many new lower bounds for M(n, d) found using these techniques for n up to 600.

Two first-order logics of permutations

We consider two orthogonal points of view on finite permutations, seen as pairs of linear orders (corresponding to the usual one line representation of permutations as words) or seen as bijections (corresponding to the algebraic point of view). For each of them, we define a corresponding first-order logical theory, that we call TOTO (Theory Of Two Orders) and TOOB (Theory Of One Bijection) respectively. We consider various expressibility questions in these theories.Our main results go in three different directions. First, we prove that, for all k≥1, the set of k-stack sortable permutations in the sense of West is expressible in TOTO, and that a logical sentence describing this set can be obtained automatically. Previously, descriptions of this set were only known for k⩽3. Next, we characterize permutation classes inside which it is possible to express in TOTO that some given points form a cycle. Lastly, we show that sets of permutations that can be described both in TOOB and TOTO are in some sense trivial. This gives a mathematical evidence that permutations-as-bijections and permutations-as-words are somewhat different objects.

Wave linked partitions and 312-avoiding permutations with primacy being 1

In this paper, we mainly study connected non-crossing linked partitions, which are vividly called wave linked partitions, and their relation to 312-avoiding permutations with primacy being 1. Let W(n) be the set of wave linked partitions of {1,2,…,n} and let w(n, k) be the number of wave linked partitions of {1,2,…,n} with wave of length k. We obtain that the enumeration of W(n+1) is counted by the well-known nth Catalan number Cn and we present the recurrence relation of w(n, k). Let P1(n) be the set of 312-avoiding permutations of {1,2,…,n} with primacy being 1. As a main result, we construct an interesting bijection between W(n) and P1(n) by introducing a labeling rule on wave linked partitions. The labeling rule implies that the major index of such kind of permutations can be easily obtained from the labels of certain vertices in the corresponding wave linked partitions.

Covering radius in the Hamming permutation space

Let Sn denote the set of permutations of {1,2,…,n}. The function f(n,s) is defined to be the minimum size of a subset S⊆Sn with the property that for any ρ∈Sn there exists some σ∈S such that the Hamming distance between ρ and σ is at most n−s. The value of f(n,2) is the subject of a conjecture by Kézdy and Snevily, which implies several famous conjectures about Latin squares. We prove that the odd n case of the Kézdy–Snevily Conjecture implies the whole conjecture. We also show that f(n,2)>3n∕4 for all n, that s!<f(n,s)<3s!(n−s)logn for 1⩽s⩽n−2 and that f(n,s)>2+2s−22n2if s⩾3.

Permutation groups arising from pattern involvement

For an arbitrary finite permutation group $G$, subgroup of the symmetric group $S_\ell$, we determine the permutations involving only members of $G$ as $\ell$-patterns, i.e., avoiding all patterns in the set $S_\ell \setminus G$. The set of all $n$-permutations with this property constitutes again a permutation group. We consequently refine and strengthen the classification of sets of permutations closed under pattern involvement and composition that is due to Atkinson and Beals.

Suitable sets of permutations, packings of triples, and Ramsey’s theorem

A set of N permutations of {1,2,…,v} is t-suitable, if each symbol precedes each subset of t−1 others in at least one permutation. The extremal problem of determining the smallest size N of such sets for given v and t was the subject of classical studies by Dushnik in 1950 and Spencer in 1971. Colbourn recently introduced the concept of suitable cores as equivalent objects of suitable sets of permutations, and studied the dual problem of determining the largest v=SCN(t,N) such that a suitable core exists for given t and N. Chan and Jedwab showed that when N=⌊t+12⌋⌈t+12⌉+l, the value of SCN (t,N) is asymptotically ⌊t2⌋+2 if l is a fixed integer. In this paper, we improve this result by showing that it is also true when l=O(lnt) using Ramsey theory. When v is bigger than ⌊t2⌋+2, we give new explicit constructions of suitable cores from packings of triples, and random constructions from extended Ramsey colorings.

Primal categories of neural polarity codes.

Neuronal membrane and synapse polarities have been attracting considerable interest in recent years. Certain functional roles for such polarities have been suggested, yet, they have largely remained a subject for speculation and debate. Here, we note that neural circuit polarity codes, defined as sets of polarity permutations, divide into primal-size circuit polarity subcodes, which, sharing certain connectivity attributes, are called categories. Two long-debated, seemingly competing paradigms of neuronal self-feedback, namely, axonal discharge and synaptic mediation, are shown to jointly define the distinction between these categories. However, as the second paradigm contains the first, it is mathematically sufficient for complete specification of all categories. The analysis of primal-size circuit polarity categories is found to reveal, explain and extend experimentally observed cortical information capacity values termed "magical numbers", associated with "working memory". While these have been previously argued on grounds of psychological experiments, here they are further supported on analytic grounds by the so-called Hebbian memory paradigm. The information dimensionality associated with these capacities is found to be a consequence of prime factorization of composite circuit polarity code sizes. Different categories of circuit polarity, identical in size and neuronal parameters, are shown to generate different firing rate dynamics.

On the topology of complexes of injective words

An injective word over a finite alphabet V is a sequence $$w=v_1v_2\cdots v_t$$ of distinct elements of V. The set $$\text {Inj}(V)$$ of injective words on V is partially ordered by inclusion. A complex of injective words is the order complex $$\Delta (W)$$ of a subposet $$W \subset \text {Inj}(V)$$. Complexes of injective words arose recently in applications of algebraic topology to neuroscience, and are of independent interest in topology and combinatorics. In this article we mainly study Permutation Complexes, i.e. complexes of injective words $$\Delta (W)$$, where W is the downward closed subposet of $$\text {Inj}(V)$$ generated by a set of permutations of V. In particular, we determine the homotopy type of $$\Delta (W)$$ when W is generated by two permutations, and prove that any stable homotopy type is realizable by a permutation complex. We describe a homotopy decomposition for the complex of injective words $$\Gamma (K)$$ associated with a simplicial complex K, and point out a connection to a result of Randal-Williams and Wahl. Finally, we discuss some probabilistic aspects of random permutation complexes.

Predicting the Time of Entry of Nanoparticles in Lipid Membranes.

The number of engineered nanoparticles for applications in the biomedical arena has grown tremendously over the last years due to advances in the science of synthesis and characterization. For most applications, the crucial step is the transport through a physiological cellular membrane. However, the behavior of nanoparticles in a biological matrix is a very complex problem that depends not only on the type of nanoparticle but also on its size, shape, phase, surface charge, chemical composition, and agglomeration state. In this paper, we introduce a streamlined theoretical model that predicts the average time of entry of nanoparticles in lipid membranes, using a combination of molecular dynamics simulations and statistical approaches. The model identifies four parameters that separate the contributions of nanoparticle characteristics (i.e., size, shape, solubility) from the membrane properties (density distribution). This factorization allows the inclusion of data obtained from both experimental and computational sources, as well as a rapid estimation of large sets of permutations in membranes. The robustness of the model is supported by experimental data carried out in lipid vesicles encapsulating graphene quantum dots as nanoparticles. Given the high level of interest across multiple areas of study in modulating intracellular targets, and the need to understand and improve the applications of nanoparticles and to assess their effect on human health (i.e., cytotoxicity, bioavailability), this work contributes to the understanding and prediction of interactions between nanoparticles and lipid membranes.

Polynomial time relatively computable triangular arrays for almost sure convergence

For 0 < x < 1, take the binary expansion with infinitely many 0's, replace each 0 with -1, this gives the polarized binary expansion of x. Let R_i(x) be the ith "polarized bit" and let S_n(x) be the sum of the first n R_i(x). {S_n} is the Z-valued random walk on (0,1). Normalize, by dividing each S_n by the square root of n: the resulting sequence converges weakly to the standard normal distribution on (0,1). The quantiles of S_n are random variables on (0,1), denoted S*_n, which are equal in distribution to the S_n, Skorokhod showed that the sequence of normalized quantiles converges almost surely to the standard normal distribution on (0,1). For n > 2, S*_n cannot be represented as the sum of the first n terms of a fixed sequence, R*_i, of random variables with the properties of the R_i. We introduce a method of constructing, for each n, an i.i.d family, R*_(n,1), ... R*_(n,n) which sums to S*_n, pointwise, as a function, not just in distribution. Each R*_(n,i) is a mean 0, variance 1 Rademacher random variable depending only on the first n bits. For each n, we get a bijection between the set of all such i.i.d. families, R*_(n,1), ... R*_(n,n), and the set of all admissible permutations of {0, ..., (2^n)-1}. Varying n, any doubly indexed such family, gives a triangular array representation of the sequence {S*_n} which is strong (because for each n, S*_n is the pointwise sum of the R*_(n,i)). Such representations are classified by sequences of admissible permutations. We show that the complexity of any sequence of admissible permutations is bounded below by that of 2^n. We explicitly construct three such polynomial time computable sequences whose complexity is bounded above by that of the function SBC (sum of binomial coefficients). We also initiate the study of some additional fine properties of admissible permutations.

Group Decision Diagram (GDD): A Compact Representation for Permutations

Permutation is a fundamental combinatorial object appeared in various areas in mathematics, computer science, and artificial intelligence. In some applications, a subset of a permutation group must be maintained efficiently. In this study, we develop a new data structure, called group decision diagram (GDD), to maintain a set of permutations. This data structure combines the zero-suppressed binary decision diagram with the computable subgroup chain of the permutation group. The data structure enables efficient operations, such as membership testing, set operations (e.g., union, intersection, and difference), and Cartesian product. Our experiments demonstrate that the data structure is efficient (i.e., 20–300 times faster) than the existing methods when the permutation group is considerably smaller than the symmetric group, or only subsets constructed by a few operations over generators are maintained.

From q-Stirling numbers to the ordered multiset partitions: A viewpoint from vincular patterns

The distribution of certain Mahonian statistic (called BAST) introduced by Babson and Steingrímsson over the set of permutations that avoid vincular pattern 132_, is shown bijectively to match the distribution of major index over the same set. This new layer of equidistribution is then applied to give alternative interpretations of two related q-Stirling numbers of the second kind, studied by Carlitz and Gould. Moreover, extensions to an Euler-Mahonian statistic over ordered set partitions, and to statistics over ordered multiset partitions present themselves naturally. The latter of which is shown to be related to the Delta Conjecture. During the course, a refined relation between BAST and its reverse complement STAT is derived as well.

A Bernstein type inequality for sums of selections from three dimensional arrays

We consider the three dimensional array A={ai,j,k}1≤i,j,k≤n, with ai,j,k∈[0,1], and the two random statistics T1≔∑i=1n∑j=1nai,j,σ(i) and T2≔∑i=1nai,σ(i),π(i), where σ and π are chosen independently from the set of permutations of {1,2,…,n}. These can be viewed as natural three dimensional generalizations of the statistic T3=∑i=1nai,σ(i), considered by Hoeffding (1951). Here we give Bernstein type concentration inequalities for T1 and T2 by extending the argument for concentration of T3 by Chatterjee (2005).

Patterns in Random Permutations Avoiding Some Sets of Multiple Patterns

We consider a random permutation drawn from the set of permutations of length n that avoid some given set of patterns of length 3. We show that the number of occurrences of another pattern sigma has a limit distribution, after suitable scaling. In several cases, the number is asymptotically normal; this contrasts to the cases of permutations avoiding a single pattern of length 3 studied in earlier papers.

Coset Partitioning Construction of Systematic Permutation Codes Under the Chebyshev Metric

The rank-modulation scheme has been recently proposed to write and store data in flash memories efficiently. In this paper, a new construction of systematic error-correcting codes for permutations is presented under the Chebyshev distance. By constructing a subgroup code and using its coset codes to partition the set of information permutations, the proposed code construction can achieve much larger code cardinality and hence higher code rates. To facilitate the encoding and decoding of the constructed codes, we also investigate the concepts of ranking and unranking for permutations, and generalize them to $M$ -ranking and $M$ -unranking for multi-permutations. Examples are provided to demonstrate the relevant concepts and the encoding/decoding algorithms.

The operators [formula omitted] on permutations, 132-avoiding permutations and inversions

We introduce the operators Fi on permutation π=π1⋯πk−11πk+1⋯πn of {1,2,…,n}, where 1≤i≤k−1, i.e., define Fi(π)=π1′π2′⋯πn′ as πj′=πj−1 for 1≤j≤i, and πi+1′πi+2′⋯πn′ has the same relative order as πi+1πi+2⋯πn. The operators Fi have many properties concerning the 132-pattern and inversions. Furthermore, we find that the operators Fi can be characterized by a series of swaps of two entries. Two applications of the operators are given. As a first application, we obtain some new objects in 132-avoiding permutations and in Dyck paths that are enumerated by the entries in Catalan’s triangle. As another application, we give an algorithm to generate the set of permutations of length n+1 with k inversions from the set of permutations of length n with k inversions when n≥k+1.

Two-machine flow shops with an optimal permutation schedule under a storage constraint

We consider two-machine flow shop scheduling problems with storage to minimize the makespan. Under a storage constraint, each job has its own size, and the total size of the job(s) being processed on the machine(s) and the job(s) waiting for the second operation cannot exceed the given capacity. It is known that the computational complexity is strongly NP-hard and in general an optimal schedule does not exist in the set of schedules with the same job sequences on two machines, referred to as a permutation set. Thus, we consider two cases such that an optimal schedule exists in the permutation set. These two cases have the special structure of processing times. In the first case, the processing times of two machines for each job are identical, while in the second case, machine 2 dominates machine 1, that is, the smallest processing time of the second group of operations is larger than or equal to the largest one of the first group of operations. We show that the first case is strongly NP-hard and admits a polynomial-time approximation scheme (PTAS), while the second case is solved in polynomial time.