Permutation Classes of Polynomial Growth

A pattern class is a set of permutations closed under the formation of subpermutations. Such classes can be characterised as those permutations not involving a particular set of forbidden permutations. A simple collection of necessary and sufficient conditions on sets of forbidden permutations which ensure that the associated pattern class is of polynomial growth is determined. A catalogue of all such sets of forbidden permutations having three or fewer elements is provided together with bounds on the degrees of the associated enumerating polynomials.Comment: 17 pages, 4 figure

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 enumeration of three pattern classes using monotone grid classes

The structure of the three pattern classes defined by the sets of forbidden permutations \{2143,4321\}, \{2143,4312\} and \{1324,4312\} is determined using the machinery of monotone grid classes. This allows the permutations in these classes to be described in terms of simple diagrams and regular languages and, using this, the rational generating functions which enumerate these classes are determined

Subclasses of the separable permutations

We prove that all subclasses of the separable permutations not containing Av(231) or a symmetry of this class have rational generating functions. Our principal tools are partial well-order, atomicity, and the theory of strongly rational permutation classes introduced here for the first time

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

Inflations of Geometric Grid Classes: Three Case Studies

We enumerate three specific permutation classes defined by two forbidden patterns of length four. The techniques involve inflations of geometric grid classes

X-ray Localization of the Globular Cluster G1 with XMM-Newton

We present an accurate X-ray position of the massive globular cluster G1 by using XMM-Newton and the Hubble Space Telescope (HST). The X-ray emission of G1 has been detected recently with XMM-Newton. There are two possibilities for the origin of the X-ray emission. It can be either due to accretion of the central intermediate-mass black hole, or by ordinary low-mass X-ray binaries. The precise location of the X-ray emission might distinguish between these two scenarios. By refining the astrometry of the XMM-Newton and HST data, we reduced the XMM-Newton error circle to 1.5". Despite the smaller error circle, the precision is not sufficient to distinguish an intermediate-mass black hole and luminous low-mass X-ray binaries. This result, however, suggests that future Chandra observations may reveal the origin of the X-ray emission.Comment: 4 pages, 2 figures; accepted for publication in Ap