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

Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs

We discover new hereditary classes of graphs that are minimal (with respect to set inclusion) of unbounded clique-width. The new examples include split permutation graphs and bichain graphs. Each of these classes is characterised by a finite list of minimal forbidden induced subgraphs. These, therefore, disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming that all such minimal classes must be defined by infinitely many forbidden induced subgraphs. In the same paper, Daligault, Rao and Thomasse make another conjecture that every hereditary class of unbounded clique-width must contain a labelled infinite antichain. We show that the two example classes we consider here satisfy this conjecture. Indeed, they each contain a canonical labelled infinite antichain, which leads us to propose a stronger conjecture: that every hereditary class of graphs that is minimal of unbounded clique-width contains a canonical labelled infinite antichain.Comment: 17 pages, 7 figure

Growth rates for subclasses of Av(321)

Pattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates

Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs

We discover new hereditary classes of graphs that are minimal (with respect to set inclusion) of unbounded clique-width. The new examples include split permutation graphs and bichain graphs. Each of these classes is characterised by a finite list of minimal forbidden induced subgraphs. These, therefore, disprove a conjecture due to Daligault, Rao and Thomassé from 2010 claiming that all such minimal classes must be defined by infinitely many forbidden induced subgraphs. In the same paper, Daligault, Rao and Thomassé make another conjecture that every hereditary class of unbounded clique-width must contain a labelled infinite antichain. We show that the two example classes we consider here satisfy this conjecture. Indeed, they each contain a canonical labelled infinite antichain, which leads us to propose a stronger conjecture: that every hereditary class of graphs that is minimal of unbounded clique-width contains a canonical labelled infinite antichain

A Counterexample Regarding Labelled Well-Quasi-Ordering

Korpelainen, Lozin, and Razgon conjectured that a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by only finitely many minimal forbidden induced subgraphs is labelled well-quasi-ordered, a notion stronger than that of n-well-quasi-order introduced by Pouzet in the 1970s. We present a counterexample to this conjecture. In fact, we exhibit a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by finitely many minimal forbidden induced subgraphs yet is not 2-well-quasi-ordered. This counterexample is based on the widdershins spiral, which has received some study in the area of permutation patterns

Governance tools for board members : adapting strategy maps and balanced scorecards for directorial action

The accountability of members of the board of directors of publicly traded companies has increased over years. Corresponding to these developments, there has been an inadequate advancement of tools and frameworks to help directorial functioning. This paper provides an argument for design of the Balanced Scorecard and Strategy Maps made available to the directors as a means of influencing, monitoring, controlling and assisting managerial action. This paper examines how the Balanced Scorecard and Strategy Maps could be modified and used for this purpose. The paper suggests incorporating Balanced Scorecards in the Internal Process perspective, ‘internal’ implying here not just ‘internal to the firm’, but also ‘internal to the inter-organizational system’. We recommend that other such factors be introduced separately under a new ‘perspective’ depending upon what the board wants to emphasize without creating any unwieldy proliferation of measures. Tracking the Strategy Map over time by the board of directors is a way for the board to take responsibility for the firm’s performance. The paper makes a distinction between action variables and monitoring variables. Monitoring variables are further divided on the basis of two considerations: a) whether results have been met or not and b) whether causative factors have met the expected levels of performance or not. Based on directorial responsibilities and accountability, we take another look at how the variables could be specified more completely and accurately with directorial recommendations for executives

Zeros of the Möbius function of permutations

We show that if a permutation \pi contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function \mu[1,\pi] of the interval [1,\pi] is zero. As a consequence, we prove that the proportion of permutations of length $\textit{n}$ with principal Möbius function equal to zero is asymptotically bounded below by (1\ -\ \sfrac{1}{e)^2} \geq 0.3995. This is the first result determining the value of \mu\left[1,\pi\right] for an asymptotically positive proportion of permutations \pi. We further establish other general conditions on a permutation \pi that ensure \mu\left[1,\pi\right]\ =\ 0, including the occurrence in \pi of any interval of the form \alpha\oplus\ 1\ \oplus\ \beta

Classification of gluteal muscle contracture in children and outcome of different treatments

<p>Abstract</p> <p>Background</p> <p>Gluteal muscle contracture (GMC) is a clinical syndrome due to multiple etiologies in which hip movements may be severely limited. The aim of this study was to propose a detailed classification of GMC and evaluate the statistical association between outcomes of different management and patient conditions.</p> <p>Methods</p> <p>One hundred fifty-eight patients, who were treated between January 1995 and December 2004, were reviewed at a mean duration of follow-up of 4.8 years. Statistical analyses were performed using X²and Fisher's exact tests.</p> <p>Results</p> <p>Non-operative management (NOM), as a primary treatment, was effective in 19 of 49 patients (38.8%), while operative management was effective in all 129 patients, with an excellence rating of 83.7% (108/129). The outcome of NOM in level I patients was significantly higher than in level II and III patients (<it>P </it>< 0.05). The results of NOM and operative management in the child group were better than the adolescent group (<it>P </it>< 0.05). Complications in level III were more than in level II.</p> <p>Conclusion</p> <p>NOM was more effective in level I patients than in level II and III patients. Operative management was effective in patients at all levels, with no statistical differences between levels or types. We recommend NOM as primary treatment for level I patients and operative management for level II and III patients. Either NOM or operative management should be carried out as early as possible.</p

Analysis of the vomeronasal organ transcriptome reveals variable gene expression depending on age and function in rabbits

The vomeronasal organ (VNO) is a chemosensory organ specialized in pheromone detection that shows a broad morphofunctional and genomic diversity among mammals. However, its expression patterns have only been well-characterized in mice. Here, we provide the first comprehensive RNA sequencing study of the rabbit VNO across gender and sexual maturation stages. We characterized the VNO transcriptome, updating the number and expression of the two main vomeronasal receptor families, including 128 V1Rs and 67 V2Rs. Further, we defined the expression of formyl-peptide receptor and transient receptor potential channel families, both known to have specific roles in the VNO. Several sex hormone-related pathways were consistently enriched in the VNO, highlighting the relevance of this organ in reproduction. Moreover, whereas juvenile and adult VNOs showed significant transcriptome differences, male and female did not. Overall, these results contribute to understand the genomic basis of behavioural responses mediated by the VNO in a non-rodent modelS