Promotion on oscillating and alternating tableaux and rotation of matchings and permutations

Using Henriques' and Kamnitzer's cactus groups, Sch\"utzenberger's promotion and evacuation operators on standard Young tableaux can be generalised in a very natural way to operators acting on highest weight words in tensor products of crystals. For the crystals corresponding to the vector representations of the symplectic groups, we show that Sundaram's map to perfect matchings intertwines promotion and rotation of the associated chord diagrams, and evacuation and reversal. We also exhibit a map with similar features for the crystals corresponding to the adjoint representations of the general linear groups. We prove these results by applying van Leeuwen's generalisation of Fomin's local rules for jeu de taquin, connected to the action of the cactus groups by Lenart, and variants of Fomin's growth diagrams for the Robinson-Schensted correspondence

Increasing and Decreasing Sequences in Fillings of Moon Polyominoes

We present an adaptation of jeu de taquin and promotion for arbitrary fillings of moon polyominoes. Using this construction we show various symmetry properties of such fillings taking into account the lengths of longest increasing and decreasing chains. In particular, we prove a conjecture of Jakob Jonsson. We also relate our construction to the one recently employed by Christian Krattenthaler, thus generalising his results.Comment: fixed typo

Some remarks on sign-balanced and maj-balanced posets

Let P be a poset with elements 1,2,...,n. We say that P is sign-balanced if exactly half the linear extensions of P (regarded as permutations of 1,2,...,n) are even permutations, i.e., have an even number of inversions. This concept first arose in the work of Frank Ruskey, who was interested in the efficient generation of all linear extensions of P. We survey a number of techniques for showing that posets are sign-balanced, and more generally, computing their "imbalance." There are close connections with domino tilings and, for certain posets, a "domino generalization" of Schur functions due to Carre and Leclerc. We also say that P is maj-balanced if exactly half the linear extensions of P have even major index. We discuss some similarities and some differences between sign-balanced and maj-balanced posets.Comment: 30 pages. Some inaccuracies in Section 3 have been corrected, and Conjecture 3.6 has been adde

Estimating Fire Weather Indices via Semantic Reasoning over Wireless Sensor Network Data Streams

Wildfires are frequent, devastating events in Australia that regularly cause significant loss of life and widespread property damage. Fire weather indices are a widely-adopted method for measuring fire danger and they play a significant role in issuing bushfire warnings and in anticipating demand for bushfire management resources. Existing systems that calculate fire weather indices are limited due to low spatial and temporal resolution. Localized wireless sensor networks, on the other hand, gather continuous sensor data measuring variables such as air temperature, relative humidity, rainfall and wind speed at high resolutions. However, using wireless sensor networks to estimate fire weather indices is a challenge due to data quality issues, lack of standard data formats and lack of agreement on thresholds and methods for calculating fire weather indices. Within the scope of this paper, we propose a standardized approach to calculating Fire Weather Indices (a.k.a. fire danger ratings) and overcome a number of the challenges by applying Semantic Web Technologies to the processing of data streams from a wireless sensor network deployed in the Springbrook region of South East Queensland. This paper describes the underlying ontologies, the semantic reasoning and the Semantic Fire Weather Index (SFWI) system that we have developed to enable domain experts to specify and adapt rules for calculating Fire Weather Indices. We also describe the Web-based mapping interface that we have developed, that enables users to improve their understanding of how fire weather indices vary over time within a particular region.Finally, we discuss our evaluation results that indicate that the proposed system outperforms state-of-the-art techniques in terms of accuracy, precision and query performance.Comment: 20pages, 12 figure

The oriented swap process and last passage percolation

We present new probabilistic and combinatorial identities relating three random processes: the oriented swap process on $n$ particles, the corner growth process, and the last passage percolation model. We prove one of the probabilistic identities, relating a random vector of last passage percolation times to its dual, using the duality between the Robinson-Schensted-Knuth and Burge correspondences. A second probabilistic identity, relating those two vectors to a vector of 'last swap times' in the oriented swap process, is conjectural. We give a computer-assisted proof of this identity for $n\le 6$ after first reformulating it as a purely combinatorial identity, and discuss its relation to the Edelman-Greene correspondence. The conjectural identity provides precise finite-$n$ and asymptotic predictions on the distribution of the absorbing time of the oriented swap process, thus conditionally solving an open problem posed by Angel, Holroyd and Romik.Comment: 36 pages, 6 figures. Full version of the FPSAC 2020 extended abstract arXiv:2003.0333