Permutation patterns and statistics

Let S_n denote the symmetric group of all permutations of the set {1, 2, ...,n} and let S = \cup_{n\ge0} S_n. If Pi is a set of permutations, then we let Av_n(Pi) be the set of permutations in S_n which avoid every permutation of Pi in the sense of pattern avoidance. One of the celebrated notions in pattern theory is that of Wilf-equivalence, where Pi and Pi' are Wilf equivalent if #Av_n(Pi)=#Av_n(Pi') for all n\ge0. In a recent paper, Sagan and Savage proposed studying a q-analogue of this concept defined as follows. Suppose st:S->N is a permutation statistic where N represents the nonnegative integers. Consider the corresponding generating function, F_n^{st}(Pi;q) = sum_{sigma in Av_n(Pi)} q^{st sigma}, and call Pi,Pi' st-Wilf equivalent if F_n^{st}(Pi;q)=F_n^{st}(Pi';q) for all n\ge0. We present the first in-depth study of this concept for the inv and maj statistics. In particular, we determine all inv- and maj-Wilf equivalences for any Pi containd in S_3. This leads us to consider various q-analogues of the Catalan numbers, Fibonacci numbers, triangular numbers, and powers of two. Our proof techniques use lattice paths, integer partitions, and Foata's fundamental bijection. We also answer a question about Mahonian pairs raised in the Sagan-Savage article.Comment: 28 pages, 5 figures, tightened up the exposition, noted that some of the conjectures have been prove

Improved Optimal and Approximate Power Graph Compression for Clearer Visualisation of Dense Graphs

Drawings of highly connected (dense) graphs can be very difficult to read. Power Graph Analysis offers an alternate way to draw a graph in which sets of nodes with common neighbours are shown grouped into modules. An edge connected to the module then implies a connection to each member of the module. Thus, the entire graph may be represented with much less clutter and without loss of detail. A recent experimental study has shown that such lossless compression of dense graphs makes it easier to follow paths. However, computing optimal power graphs is difficult. In this paper, we show that computing the optimal power-graph with only one module is NP-hard and therefore likely NP-hard in the general case. We give an ILP model for power graph computation and discuss why ILP and CP techniques are poorly suited to the problem. Instead, we are able to find optimal solutions much more quickly using a custom search method. We also show how to restrict this type of search to allow only limited back-tracking to provide a heuristic that has better speed and better results than previously known heuristics.Comment: Extended technical report accompanying the PacificVis 2013 paper of the same nam

Interactive, Constraint-based Layout of Engineering Diagrams

Many engineering disciplines require visualisation of networks. Constrained graph layout is a powerful new approach to network layout that allows the user to impose a wide variety application-specific placement constraints—such as downwards pointing directed edges, alignment of nodes, cluster containment and non-overlapping nodes and clusters—on the layout. We have recently developed an efficient algorithm for topology-preserving constrained graph layout. This underpins two dynamic graph layout applications we have developed: a network diagram authoring tool, Dunnart, and a network diagram browser. In this paper we provide an overview of topology-preserving constrained graph layout and illustrate how Dunnart and the network diagram browser can be applied to engineering diagram authoring and visualisation

Extending adjacency matrices to 3D with triangles

Social networks are the fabric of society and the subject of frequent visual analysis. Closed triads represent triangular relationships between three people in a social network and are significant for understanding inherent interconnections and influence within the network. The most common methods for representing social networks (node-link diagrams and adjacency matrices) are not optimal for understanding triangles. We propose extending the adjacency matrix form to 3D for better visualization of network triads. We design a 3D matrix reordering technique and implement an immersive interactive system to assist in visualizing and analyzing closed triads in social networks. A user study and usage scenarios demonstrate that our method provides substantial added value over node-link diagrams in improving the efficiency and accuracy of manipulating and understanding the social network triads.Comment: 10 pages, 8 figures and 3 table

Psychosis and Human Rights: Conflicts in Mental Health Policy and Practice

This paper examines conflicts in polices in England and Wales pertaining to the demand for alternative, non-medical crisis support for those experiencing ‘psychosis’. We examine the limitations of current treatment, policy and legislative frameworks in supporting these demands. In particular, we focus on the limitations of prevailing conceptualisations of ‘human rights’, ‘social inclusion’ and ‘recovery’. These concepts, we argue, are embedded within a broader treatment framework which renders medication as mandatory and all other treatment modalities as inherently subsidiary, and a broader policy framework which is complicit with bio-medical orthodoxies of ‘mental illness’ and prioritises treatment compliance and compulsion. Therefore, in order to advance a ‘human rights’ approach to mental health policy, we argue that reigning orthodoxies inherent within policy and practice must be explicitly challenged to open up spaces for the availability of alternatives

Hanstreamer: an Open-source Webcam-based Live Data Presentation System

We present Hanstreamer, a free and open-source system for webcam-based data presentation. The system performs real-time gesture recognition on the user's webcam video stream to provide interactive data visuals. Apart from the standard chart and map visuals, Hanstreamer is the first such video data presentation system to support network visualisation and interactive DimpVis-style time-series data exploration. The system is ready for use with popular online meeting software such as Zoom and Microsoft Teams.Comment: 3 pages, 5 figure