2-stack pushall sortable permutations

In the 60's, Knuth introduced stack-sorting and serial compositions of stacks. In particular, one significant question arise out of the work of Knuth: how to decide efficiently if a given permutation is sortable with 2 stacks in series? Whether this problem is polynomial or NP-complete is still unanswered yet. In this article we introduce 2-stack pushall permutations which form a subclass of 2-stack sortable permutations and show that these two classes are closely related. Moreover, we give an optimal O(n^2) algorithm to decide if a given permutation of size n is 2-stack pushall sortable and describe all its sortings. This result is a step to the solve the general 2-stack sorting problem in polynomial time.Comment: 41 page

Sorting with a forklift

A fork stack is a generalised stack which allows pushes and pops of several items at a time. We consider the problem of determining which input streams can be sorted using a single forkstack, or dually, which permutations of a fixed input stream can be produced using a single forkstack. An algorithm is given to solve the sorting problem and the minimal unsortable sequences are found. The results are extended to fork stacks where there are bounds on how many items can be pushed and popped at one time. In this context we also establish how to enumerate the collection of sortable sequences.Comment: 24 pages, 2 figure

The Matrix Sortability Problem

Sorting is such a fundamental component of achieving efficiency that a significant body of mathematics is dedicated to the investigation of sorting. Any modern textbook on algorithms contains chapters on sorting. One approach to arranging a disorganized list of items into an organized list is to successively identify two blocks of contiguous items, and swap the two blocks. In a fundamental paper D.A. Christie showed that a special version of block swapping, in recent times called context directed swapping and abbreviated cds, is the most efficient among block swapping strategies to achieve an organized list of items. The cds sorting strategy is also the most robust among block swap based sorting methods. It has been discovered that the context directed block swap operation on a list of objects generalizes to an operation on simple graphs. In turn it has been discovered that this operation on simple graphs corresponds with an operation on the adjacency matrix of a simple graph. The adjacency matrix is a symmetric square matrix with entries 0 and 1, and all diagonal entries 0. The corresponding operation is denoted Mcds, abbreviating matrix context directed swap. The operation on the adjacency matrix naturally employs the arithmetic of GF(2), the finite field of two elements. It has been speculated that the Mcds operation on these specific matrices over GF(2) corresponds with the more than a century old Schur complement operation on these matrices. In this thesis, we confirm this prior speculation about the correspondence between Mcds and the Schur complement, in the context of GF(2). We generalize the Mcds operation to not necessarily square matrices over arbitrary fields and we prove that the generalized Mcds corresponds with the Schur complement also in the more general context of all fields

Shard polytopes

For any lattice congruence of the weak order on permutations, N. Reading proved that gluing together the cones of the braid fan that belong to the same congruence class defines a complete fan, called a quotient fan, and V. Pilaud and F. Santos showed that it is the normal fan of a polytope, called a quotientope. In this paper, we provide a simpler approach to realize quotient fans based on Minkowski sums of elementary polytopes, called shard polytopes, which have remarkable combinatorial and geometric properties. In contrast to the original construction of quotientopes, this Minkowski sum approach extends to type $B$.Comment: 73 pages, 35 figures; Version 2: minor corrections for final versio

Do not Trash the Incentive! Monetary incentives and waste sorting

This paper examines whether monetary incentives are an effective tool for increasing domestic waste sorting. We exploit the exogenous variation in the waste management policies experienced during the years 1999-2008 by the 95 municipalities in the district of Treviso (Italy). We estimate with a panel analysis that pay-as-you-throw (PAYT) incentive schemes increase by 12.3% the sorted-total waste ratio. This increase reflects a change in the behavior of households, who keep unaltered the production of total waste but sort it to a larger extent. Our data show that household behavior is also influenced by the policies of adjacent municipalities.Incentives, environment, waste management, PAYT

Online detection and sorting of extracellularly recorded action potentials in human medial temporal lobe recordings, in vivo

Understanding the function of complex cortical circuits requires the simultaneous recording of action potentials from many neurons in awake and behaving animals. Practically, this can be achieved by extracellularly recording from multiple brain sites using single wire electrodes. However, in densely packed neural structures such as the human hippocampus, a single electrode can record the activity of multiple neurons. Thus, analytic techniques that differentiate action potentials of different neurons are required. Offline spike sorting approaches are currently used to detect and sort action potentials after finishing the experiment. Because the opportunities to record from the human brain are relatively rare, it is desirable to analyze large numbers of simultaneous recordings quickly using online sorting and detection algorithms. In this way, the experiment can be optimized for the particular response properties of the recorded neurons. Here we present and evaluate a method that is capable of detecting and sorting extracellular single-wire recordings in realtime. We demonstrate the utility of the method by applying it to an extensive data set we acquired from chronically-implanted depth electrodes in the hippocampus of human epilepsy patients. This dataset is particularly challenging because it was recorded in a noisy clinical environment. This method will allow the development of closed-loop experiments, which immediately adapt the experimental stimuli and/or tasks to the neural response observed.Comment: 9 figures, 2 tables. Journal of Neuroscience Methods 2006 (in press). Journal of Neuroscience Methods, 2006 (in press