Set Of Permutations Research Articles

Coding for locality in reconstructing permutations

The problem of storing permutations in a distributed manner arises in several common scenarios, such as efficient updates of a large, encrypted, or compressed data set. This problem may be addressed in either a combinatorial or a coding approach. The former approach boils down to presenting large sets of permutations with locality, that is, any symbol of the permutation can be computed from a small set of other symbols. In the latter approach, a permutation may be coded in order to achieve locality. Both approaches must present low query complexity to allow the user to find an element efficiently. We discuss both approaches, and give a particular focus to the combinatorial one. In the combinatorial approach, we provide upper and lower bounds for the maximal size of a set of permutations with locality, and provide several simple constructions which attain the upper bound. In cases where the upper bound is not attained, we provide alternative constructions using a variety of tools, such as Reed-Solomon codes, permutation polynomials, and multi-permutations. In addition, several low-rate constructions of particular interest are discussed. In the coding approach we discuss an alternative representation of permutations, present a paradigm for supporting arbitrary powers of the stored permutation, and conclude with a proof of concept that permutations may be stored more efficiently than ordinary strings over the same alphabet.

On rotated Schur-positive sets

The problem of finding Schur-positive sets of permutations, originally posed by Gessel and Reutenauer, has seen some recent developments. Schur-positive sets of pattern-avoiding permutations have been found by Sagan et al. and a general construction based on geometric operations on grid classes has been given by the authors. In this paper we prove that horizontal rotations of Schur-positive subsets of permutations are always Schur-positive. The proof applies a cyclic action on standard Young tableaux of certain skew shapes and a jeu-de-taquin type straightening algorithm. As a consequence of the proof we obtain a notion of cyclic descent set on these tableaux, which is rotated by the cyclic action on them.

Fast algorithms for finding pattern avoiders and counting pattern occurrences in permutations

Given a set $\Pi$ of permutation patterns of length at most $k$, we present an algorithm for building $S_{\le n}(\Pi)$, the set of permutations of length at most $n$ avoiding the patterns in $\Pi$, in time $O(|S_{\le n - 1}(\Pi)| \cdot k + |S_{n}(\Pi)|)$. Additionally, we present an $O(n!k)$-time algorithm for counting the number of copies of patterns from $\Pi$ in each permutation in $S_n$. Surprisingly, when $|\Pi| = 1$, this runtime can be improved to $O(n!)$, spending only constant time per permutation. Whereas the previous best algorithms, based on generate-and-check, take exponential time per permutation analyzed, all of our algorithms take time at most polynomial per outputted permutation. If we want to solve only the enumerative variant of each problem, computing $|S_{\le n}(\Pi)|$ or tallying permutations according to $\Pi$-patterns, rather than to store information about every permutation, then all of our algorithms can be implemented in $O(n^{k+1}k)$ space. Using our algorithms, we generated $|S_5(\Pi)|, \ldots, |S_{16}(\Pi)|$ for each $\Pi \subseteq S_4$ with $|\Pi| > 4$, and analyzed OEIS matches. We obtained a number of potentially novel pattern-avoidance conjectures. Our algorithms extend to considering permutations in any set closed under standardization of subsequences. Our algorithms also partially adapt to considering vincular patterns.

$G$ method in action: Fast exact sampling from set of permutations of order $n$ according to Mallows model through Cayley metric

Using $G$ method, we give a fast exact (not approximate) Markovian method for sampling from $\mathbb{S}_{n}$, the set of permutations of order $n$, according to the Mallows model through Cayley metric (a model for ranked data). This method has something in common with the cyclic Gibbs sampler and something in common with the swapping method. The number of steps of our method is equal to the number of steps of swapping method, that is, $n-1$; moreover, both methods use the best probability distributions on sampling, the swapping method uses uniform probability distributions while our method uses almost uniform probability distributions (all the components of an almost uniform probability distribution are, here, identical, excepting at most one of them). But, besides sampling, we can do other things for the Mallows model through Cayley metric—we compute the normalizing constant and, by Uniqueness theorem, certain important probabilities.

An algorithm computing combinatorial specifications of permutation classes

This article presents a methodology that automatically derives a combinatorial specification for a permutation class C, given its basis B of excluded patterns and the set of simple permutations in C, when these sets are both finite. This is achieved considering both pattern avoidance and pattern containment constraints in permutations. The obtained specification yields a system of equations satisfied by the generating function of C, this system being always positive and algebraic. It also yields a uniform random sampler of permutations in C. The method presented is fully algorithmic.

A permutation code preserving a double Eulerian bistatistic

In 1977 Foata proved bijectively, among other things, that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations, and the generating function of the corresponding bistatistics is the double Eulerian polynomial. In 2013 Foata’s result was rediscovered by Visontai as a conjecture, and then reproved by Aas in 2014.In this paper, we define a permutation code (that is, a bijection between permutations and subexcedant sequences) and show the more general result that two 5-tuples of set-valued statistics on the set of permutations and on the set of subexcedant sequences, respectively, are equidistributed. In particular, these results give another bijective proof of Foata’s result.

Largest Minimal Inversion-Complete and Pair-Complete Sets of Permutations

We solve two related extremal problems in the theory of permutations. A set 𝑄 of permutations of the integers 1 to 𝑛 is inversion-complete (resp., pair-complete) if for every inversion (𝑗, 𝑖), where 1 ≤ 𝑖 < 𝑗 ≤ 𝑛, (resp., for every pair (𝑖, 𝑗), where 𝑖 ≠ 𝑗) there exists a permutation in 𝑄 where 𝑗 is before 𝑖. It is minimally inversion-complete if in addition no proper subset of 𝑄 is inversion-complete; and similarly for pair completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion-complete set of permutations, and that of a minimal pair-complete set of permutations. The latter problem arises in the determination of the Caratheodory numbers for certain abstract convexity structures on the (𝑛 − 1)-dimensional real and integer vector spaces. Using Mantel's Theorem on the maximum number of edges in a triangle-free graph, we determine these two maximum cardinalities and we present a complete description of the optimal sets of permutations for each problem. Perhaps surprisingly (since there are twice as many pairs to cover as inversions), these two maximum cardinalities coincide whenever 𝑛 ≥ 4.

Right-jumps and pattern avoiding permutations

We study the iteration of the process "a particle jumps to the right" in permutations. We prove that the set of permutations obtained in this model after a given number of iterations from the identity is a class of pattern avoiding permutations. We characterize the elements of the basis of this class and we enumerate these "forbidden minimal patterns" by giving their bivariate exponential generating function: we achieve this via a catalytic variable, the number of left-to-right maxima. We show that this generating function is a D-finite function satisfying a nice differential equation of order~2. We give some congruence properties for the coefficients of this generating function, and we show that their asymptotics involves a rather unusual algebraic exponent (the golden ratio $(1+\sqrt 5)/2$) and some unusual closed-form constants. We end by proving a limit law: a forbidden pattern of length $n$ has typically $(\ln n) /\sqrt{5}$ left-to-right maxima, with Gaussian fluctuations.

Decomposing inversion sets of permutations and applications to faces of the Littlewood–Richardson cone

If \(\alpha \in S_n\) is a permutation of \(\{1, 2,\ldots ,n\}\), the inversion set of \(\alpha \) is \(\Phi (\alpha ) = \{ (i, j) \,| \, 1 \leqslant i < j \leqslant n, \alpha (i) > \alpha (j)\}\). We describe all r-tuples \(\alpha _1, \alpha _2, \ldots , \alpha _r \in S_n\) such that \(\Delta _n^+ = \{ (i, j) \, | \, 1 \leqslant i < j \leqslant n\}\) is the disjoint union of \(\Phi (\alpha _1), \Phi (\alpha _2), \ldots , \Phi (\alpha _r)\). Using this description, we prove that certain faces of the Littlewood–Richardson cone are simplicial and provide an algorithm for writing down their sets of generating rays. We also discuss analogous problems for the Weyl groups of root systems of types B, C and D providing solutions for types B and C. Finally, we provide some enumerative results and introduce a useful tool for visualizing inversion sets.

Constructions and nonexistence results for suitable sets of permutations

A set of N permutations of {1,2,…,v} is (N,v,t)-suitable if each symbol precedes each subset of t−1 others in at least one permutation. The central problems are to determine the smallest N for which such a set exists for given v and t, and to determine the largest v for which such a set exists for given N and t. These extremal problems were the subject of classical studies by Dushnik in 1950 and Spencer in 1971. We give examples of suitable sets of permutations for new parameter triples (N,v,t). We relate certain suitable sets of permutations with parameter t to others with parameter t+1, thereby showing that one of the two infinite families recently presented by Colbourn can be constructed directly from the other. We prove an exact nonexistence result for suitable sets of permutations using elementary combinatorial arguments. We then establish an asymptotic nonexistence result using Ramsey's theorem.

Random Permutations with Prime Lengths of Cycles

A set of $n$th order permutations with prime lengths of cycles is considered. An asymptotic estimate for the number of all such permutations is obtained as $n\to\infty.$ Given a uniform distribution on the set of such permutations of order $n$, a local limit theorem is proved, evaluating the distribution of the number of cycles $\nu_n$ in a permutation selected at random. This theorem implies, in particular, that the random variable $\nu_n$ is asymptotically normal with parameters ($\log\log n$, $\log\log n$) as $n\to\infty$. It is shown that the random variable $\nu_n(p)$, the number of cycles of a fixed length $p$ in such a permutation ($p$ is a prime number), has in the limit a Poisson distribution with parameter ${1}/{p}.$ Assuming that a permutation of order $n$ is selected in accordance with the uniform distribution from the set of all such permutations with prime cycle lengths, each of which has exactly $N$ cycles $(1\le N\le[{n}/{2}]),$ limit theorems are proved, evaluating the distribution of the...

A Dimension Distance-Based SCMA Codebook Design

For sparse code multiple access (SCMA) with traditional codebooks, the initial information of message passing algorithm (MPA) receiver is easily susceptible to noise and multipath fading, and the convergence reliability of the first detected user in each decision process is unsatisfactory. Driven by these problems, an optimized codebook design for SCMA is presented in this paper. In the proposed SCMA codebook design, we first use turbo trellis coded modulation technology to design a basic complex multi-dimension constellation, which can increase the minimum Euclidean distance. Then, phase rotation and coordinate interleaving are added on the constellation to increase diversity and coordinate product distance between any constellation points. Based on these, we propose a novel criterion to select the most appropriate permutation set, which can capture as large as the sum of distance between dimensions of interfering codewords multiplexed on each resource node and maximize the diversity over the set of the sums of distance between dimensions of interfering codewords multiplexed on all resource nodes. Benefiting from the proposed codebook design, the quality of initial information of MPA receiver on each resource node and the convergence reliability of the first detected user in each decision process will be improved. Simulation results show that the bit error rate performance of SCMA with the proposed codebooks outperforms SCMA with traditional codebooks, low-density signature, and orthogonal frequency division multiple access under the same load.

Special Transpositions of Permutation Elements and Properties of Their Composition

We propose a strategy to solve the problem of optimization of a linear function on the set of cyclic permutations. The strategy is based on the properties of transpositions of special kind. The properties of this class of transpositions are investigated. Assertions about the impact of compositions of such transpositions on an arbitrary permutation are proved. For the approximate solutions obtained using the above strategy, estimation is substantiated.

UNDECIDABILITY AND THE DEVELOPABILITY OF PERMUTOIDS AND RIGID PSEUDOGROUPS

A permutoid is a set of partial permutations that contains the identity and is such that partial compositions, when defined, have at most one extension in the set. In 2004 Peter Cameron conjectured that there can exist no algorithm that determines whether or not a permutoid based on a finite set can be completed to a finite permutation group. In this note we prove Cameron’s conjecture by relating it to our recent work on the profinite triviality problem for finitely presented groups. We also prove that the existence problem for finite developments of rigid pseudogroups is unsolvable. In an appendix, Steinberg recasts these results in terms of inverse semigroups.

Spaceability in norm-attaining sets

We study the existence of infinite-dimensional vector spaces in the sets of norm-attaining operators, multilinear forms, and polynomials. Our main result is that, for every set of permutations P of the set {1,…,n}, there exists a closed infinite-dimensional Banach subspace of the space of n-linear forms on ℓ1 such that, for all nonzero elements B of such a subspace, the Arens extension associated to the permutation σ of B is norm-attaining if and only if σ is an element of P. We also study the structure of the set of norm-attaining n-linear forms on c0.

On the exact solution of the no-wait flow shop problem with due date constraints

This paper deals with the no-wait flow shop scheduling problem with due date constraints. In the no-wait flow shop problem, waiting time is not allowed between successive operations of jobs. Moreover, the jobs should be completed before their respective due dates; due date constraints are dealt with as hard constraints. The considered performance criterion is makespan. The problem is strongly NP-hard. This paper develops a number of distinct mathematical models for the problem based on different decision variables. Namely, a mixed integer programming model, two quadratic mixed integer programming models, and two constraint programming models are developed. Moreover, a novel graph representation is developed for the problem. This new modeling technique facilitates the investigation of some of the important characteristics of the problem; this results in a number of propositions to rule out a large number of infeasible solutions from the set of all possible permutations. Afterward, the new graph representation and the resulting propositions are incorporated into a new exact algorithm to solve the problem to optimality. To investigate the performance of the mathematical models and to compare them with the developed exact algorithm, a number of test problems are solved and the results are reported. Computational results demonstrate that the developed algorithm is significantly faster than the mathematical models.

Systematic Generation, Analysis, and Characterization of 3D Micro-architected Metamaterials

Controlling the unit-cell topology of microlattice structures can enable the customization of effective anisotropic material properties. A wide range of properties can be obtained by varying connectivity within the unit cell, which then can be further used to optimize structures specific to applications. A methodology for a systematic generation of microlattice structures is presented that focuses on controlling discrete topology instead of average porosity (as is done in conventional porous media). An algorithm is developed to create valid lattice structures without redundancies from a given set of template nodes. A set of possible permutations of structures from an eight-node cubic octant of a unit cell are generated for evaluation of the degree of anisotropy. Generic models are developed to calculate the effective thermal and mechanical properties as an effect of topology and porosity of the micro-architected structure. The thermal and mechanical anisotropies are investigated for the effective properties of micro-architected materials. A few of the structured materials are fabricated using 3D printing technology and their effective properties characterized. Structures are represented as graphs in the form of adjacency matrices. Effective thermal conductivity is analyzed using a resistance network model, and effective stiffness is evaluated using a self-consistent elastic model, respectively. A total of 160 000 structures are generated and compared to porous-metal foams in which porosity is one of the design variables. The results show that it is possible to obtain a wide range of properties spanning more than an order of magnitude in comparison to porous-metal structures. Structures with a maximum anisotropy ratios of 7.1 and 8.2 are observed for thermal and mechanical properties, respectively. Preliminary experimental results validated the anisotropy ratio for the thermal conductivity and stiffness.

Graph-intersecting set systems and LYM inequalities

For a graph H, an H-intersecting collection is a collection of packings on a ground set [n] with V(H) parts where, for every edge ab of H, every two distinct packings in the collection, P1 and P2, satisfy P1a∩P2b≠Ø. We give a general method for producing LYM-like inequalities for H-intersecting collections, for any graph H. Our technique involves counting permutations having a certain “following” pattern of sets from the collection. For every graph H, we provide the optimal set of permutations to count, encoded by the arcs of a “follow digraph” F. We fully characterize the relationship between graphs H and follow digraphs F. Our inequalities inherently bound the maximum number of packings in H-intersecting collections, and for specific graphs H, we use them to derive bounds. If H is the star on 3 vertices, then an H-intersecting collection has size at most 12a+ba=O(φn∕n), where φ=12(1+5), a≈φb, and a+2b=n. If H is Kv with a loop on each vertex, with v≥3, then an H-intersecting collection has size at most 14⌊2n∕v⌋⌊n∕v⌋, for all n≥22v∕3, improving a bound given by Poljak and Tuza by a factor of 2. Assuming that optimal H-intersecting collections have a limiting behaviour, we derive a simplified form for our inequalities. When H is a complete graph Kv with a loop on each vertex, then, under these assumptions, we prove that an H-intersecting collection has size at most 1v2−v2n∕v+2n∕v+1(1+o(1)), which improves our bound for large enough n and v.

Continuous Representations and Functional Extensions in Combinatorial Optimization

The concepts of functional representation of a set of points of the Euclidean arithmetic space and an extension of functions from the set onto its superset are introduced. Functional representations of sets are related to their extensions. Strict functional representations of the Boolean set, general permutation, and polypermutation sets are derived. The advantages of applying strict representations of Euclidean combinatorial sets to construct functional extensions from them and to solve combinatorial problems are presented.

Frankl-Rödl-type theorems for codes and permutations

We give a new proof of the Frankl-R\"odl theorem on forbidden intersections, via the probabilistic method of dependent random choice. Our method extends to codes with forbidden distances, where over large alphabets our bound is significantly better than that obtained by Frankl and R\"odl. We also apply our bound to a question of Ellis on sets of permutations with forbidden distances, and to establish a weak form of a conjecture of Alon, Shpilka and Umans on sunflowers.