Universal layered permutations
We establish an exact formula for the length of the shortest permutation containing all layered permutations of length $n$, proving a conjecture of Gray.
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arXiv subjects
Michael Albert
Publications and source records attributed to Michael Albert.
An elementary proof of Bevan's theorem on the growth of grid classes of permutations
Bevan established that the growth rate of a monotone grid class of permutations is equal to the square of the spectral radius of a related bipartite graph. We give an elementary and self-contained proof of a generalization of this result using only Stirling's Formula, the method of Lagrange multipliers, and the singular value decomposition of matrices.
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Traffic Optimization For a Mixture of Self-interested and Compliant Agents
This paper focuses on two commonly used path assignment policies for agents traversing a congested network: self-interested routing, and system-optimum routing. In the self-interested routing policy each agent selects a path that optimizes its own utility, while the system-optimum routing agents are assigned paths with the goal of maximizing system performance. This paper considers a scenario where a centralized network manager wishes to optimize utilities over all agents, i.e., implement a system-optimum routing policy. In many real-life scenarios, however, the system manager is unable to influence the route assignment of all agents due to limited influence on route choice decisions. Motivated by such scenarios, a computationally tractable method is presented that computes the minimal amount of agents that the system manager needs to influence (compliant agents) in order to achieve system optimal performance. Moreover, this methodology can also determine whether a given set of compliant agents is sufficient to achieve system optimum and compute the optimal route assignment for the compliant agents to do so. Experimental results are presented showing that in several large-scale, realistic traffic networks optimal flow can be achieved with as low as 13% of the agent being compliant and up to 54%.
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Complexity of Scheduling Charging in the Smart Grid
In the smart grid, the intent is to use flexibility in demand, both to balance demand and supply as well as to resolve potential congestion. A first prominent example of such flexible demand is the charging of electric vehicles, which do not necessarily need to be charged as soon as they are plugged in. The problem of optimally scheduling the charging demand of electric vehicles within the constraints of the electricity infrastructure is called the charge scheduling problem. The models of the charging speed, horizon, and charging demand determine the computational complexity of the charge scheduling problem. For about 20 variants, we show, using a dynamic programming approach, that the problem is either in P or weakly NP-hard. We also show that about 10 variants of the problem are strongly NP-hard, presenting a potentially significant obstacle to their use in practical situations of scale.
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Characterising inflations of monotone grid classes of permutations
We characterise those permutation classes whose simple permutations are monotone griddable. This characterisation is obtained by identifying a set of nine substructures, at least one of which must occur in any simple permutation containing a long sum of 21s.
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On the growth of merges and staircases of permutation classes
There is a well-known upper bound on the growth rate of the merge of two permutation classes. Curiously, there is no known merge for which this bound is not achieved. Using staircases of permutation classes, we provide sufficient conditions for this upper bound to be achieved. In particular, our results apply to all merges of principal permutation classes. We end by demonstrating how our techniques can be used to reprove a result of Bóna.
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Unsplittable classes of separable permutations
A permutation class is splittable if it is contained in the merge of two of its proper subclasses. We characterise the unsplittable subclasses of the class of separable permutations both structurally and in terms of their bases.
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Auto-JacoBin: Auto-encoder Jacobian Binary Hashing
Binary codes can be used to speed up nearest neighbor search tasks in large scale data sets as they are efficient for both storage and retrieval. In this paper, we propose a robust auto-encoder model that preserves the geometric relationships of high-dimensional data sets in Hamming space. This is done by considering a noise-removing function in a region surrounding the manifold where the training data points lie. This function is defined with the property that it projects the data points near the manifold into the manifold wisely, and we approximate this function by its first order approximation. Experimental results show that the proposed method achieves better than state-of-the-art results on three large scale high dimensional data sets.
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$2\times 2$ monotone grid classes are finitely based
In this note, we prove that all $2 \times 2$ monotone grid classes are finitely based, i.e., defined by a finite collection of minimal forbidden permutations. This follows from a slightly more general result about certain $2 \times 2$ (generalized) grid classes having two monotone cells in the same row.
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Collatz meets Fibonacci
The Collatz map is defined for a positive even integer as half that integer, and for a positive odd integer as that integer threefold, plus one. The Collatz conjecture states that when the map is iterated the number one is eventually reached. We study permutations that arise as sequences from this iteration. We show that permutations of this type of length up to 14 are enumerated by the Fibonacci numbers. Beyond that excess permutations appear. We will explain the appearance of these excess permutations and give an upper bound on the exact enumeration.
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Equipopularity Classes in the Separable Permutations
When two patterns occur equally often in a set of permutations, we say that these patterns are equipopular. Using both structural and analytic tools, we classify the equipopular patterns in the set of separable permutations. In particular, we show that the number of equipopularity classes for length $n$ patterns in the separable permutations is equal to the number of partitions of $n-1$.
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Permutations sortable by two stacks in parallel and quarter plane walks
At the end of the 1960s, Knuth characterised the permutations that can be sorted using a stack in terms of forbidden patterns. He also showed that they are in bijection with Dyck paths and thus counted by the Catalan numbers. Subsequently, Even \& Itai, Pratt and Tarjan studied permutations that can be sorted using two stacks in parallel. This problem is significantly harder. In particular, a sortable permutation can now be sorted by several distinct sequences of stack operations. Moreover, in order to be sortable, a permutation must avoid infinitely many patterns. The associated counting question has remained open for 40 years. We solve it by giving a pair of functional equations that characterise the generating function of permutations that can be sorted with two parallel stacks. The first component of this system describes the generating function Q(a,u) of square lattice loops confined to the positive quadrant, counted by the length and the number of North-West and East-South factors. Our analysis of the asymptotic number of sortable permutations relies at the moment on two intriguing conjectures dealing with the series Q(a,u). We prove that they hold for loops confined to the upper half plane, or not confined at all. They remain open for quarter plane loops. Given the recent activity on walks confined to cones, we believe them to be attractive per se.
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A general theory of Wilf-equivalence for Catalan structures
The existence of apparently coincidental equalities (also called Wilf-equivalences) between the enumeration sequences, or generating functions, of various hereditary classes of combinatorial structures has attracted significant interest. We investigate such coincidences among non-crossing matchings and a variety of other Catalan structures including Dyck paths, 231-avoiding permutations and plane forests. In particular we consider principal classes defined by not containing an occurrence of a single given structure. An easily computed equivalence relation among structures is described such that if two structures are equivalent then the associated principal classes have the same enumeration sequence. We give an asymptotic estimate of the number of equivalence classes of this relation among structures of a given size and show that it is exponentially smaller than the corresponding Catalan number. In other words these "coincidental" equalities are in fact very common among principal classes. Our results also allow us to prove, in a unified and bijective manner, several known Wilf-equivalences from the literature.
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Operators of equivalent sorting power and related Wilf-equivalences
We study sorting operators $\mathbf{A}$ on permutations that are obtained composing Knuth's stack sorting operator $\mathbf{S}$ and the reversal operator $\mathbf{R}$, as many times as desired. For any such operator $\mathbf{A}$, we provide a size-preserving bijection between the set of permutations sorted by $\mathbf{S} \circ \mathbf{A}$ and the set of those sorted by $\mathbf{S} \circ \mathbf{R} \circ \mathbf{A}$, proving that these sets are enumerated by the same sequence, but also that many classical permutation statistics are equidistributed across these two sets. The description of this family of bijections is based on a bijection between the set of permutations avoiding the pattern $231$ and the set of those avoiding $132$ which preserves many permutation statistics. We also present other properties of this bijection, in particular for finding pairs of Wilf-equivalent permutation classes.
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Isomorphisms between pattern classes
Isomorphisms p between pattern classes A and B are considered. It is shown that, if p is not a symmetry of the entire set of permutations, then, to within symmetry, A is a subset of one a small set of pattern classes whose structure, including their enumeration, is determined.
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Young classes of permutations
We characterise those classes of permutations having the property that for every tableau shape either every permutation of that shape or no permutation of that shape belongs to the class. The characterisation is in terms of the dominance order for partitions (and their conjugates) and shows that for any such class there is a constant k such that no permutation in the class can contain both an increasing and a decreasing sequence of length k.
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Pattern classes and priority queues
When a set of permutations comprising a pattern class C is submitted as input to a priority queue the resulting output is again a pattern class C'. The basis of C' is determined for pattern classes C whose basis elements have length 3, and is finite in these cases. An example is given of a class C with basis 2431 for which C is not finitely based.
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The enumeration of permutations avoiding 2143 and 4231
We enumerate the pattern class Av(2143,4231) and completely describe its permutations. The main tools are simple permutations and monotone grid classes.
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