Smith Normal Form of a Multivariate Matrix Associated with Partitions

Consideration of a question of E. R. Berlekamp led Carlitz, Roselle, and Scoville to give a combinatorial interpretation of the entries of certain matrices of determinant~1 in terms of lattice paths. Here we generalize this result by refining the matrix entries to be multivariate polynomials, and by determining not only the determinant but also the Smith normal form of these matrices. A priori the Smith form need not exist but its existence follows from the explicit computation. It will be more convenient for us to state our results in terms of partitions rather than lattice paths.Comment: 12 pages; revised version (minor changes on first version); to appear in J. Algebraic Combinatoric

A Simple Approach to Error Reconciliation in Quantum Key Distribution

We discuss the error reconciliation phase in quantum key distribution (QKD) and analyse a simple scheme in which blocks with bad parity (that is, blocks containing an odd number of errors) are discarded. We predict the performance of this scheme and show, using a simulation, that the prediction is accurate.Comment: 19 pages. Presented at the 53rd Annual Meeting of the Australian Mathematical Society, Adelaide, Oct 1, 2009. See also http://wwwmaths.anu.edu.au/~brent/pub/pub239.htm

The Descent Set and Connectivity Set of a Permutation

The descent set D(w) of a permutation w of 1,2,...,n is a standard and well-studied statistic. We introduce a new statistic, the connectivity set C(w), and show that it is a kind of dual object to D(w). The duality is stated in terms of the inverse of a matrix that records the joint distribution of D(w) and C(w). We also give a variation involving permutations of a multiset and a q-analogue that keeps track of the number of inversions of w.Comment: 12 page

Mechanisms for the generation and regulation of sequential behaviour

A critical aspect of much human behaviour is the generation and regulation of sequential activities. Such behaviour is seen in both naturalistic settings such as routine action and language production and laboratory tasks such as serial recall and many reaction time experiments. There are a variety of computational mechanisms that may support the generation and regulation of sequential behaviours, ranging from those underlying Turing machines to those employed by recurrent connectionist networks. This paper surveys a range of such mechanisms, together with a range of empirical phenomena related to human sequential behaviour. It is argued that the empirical phenomena pose difficulties for most sequencing mechanisms, but that converging evidence from behavioural flexibility, error data arising from when the system is stressed or when it is damaged following brain injury, and between-trial effects in reaction time tasks, point to a hybrid symbolic activation-based mechanism for the generation and regulation of sequential behaviour. Some implications of this view for the nature of mental computation are highlighted

A study of the vegetation of the South Burn valley, Waldridge Fell, County Durham

The vegetation of the South Burn Valley, Waldridge Fell, County Durham, was examined using an objective, quantitative phytosociological method. The vegetation units found to be present in the valley using this technique were then described and classified. Certain parameters relating to the chemical and physical environment of each vegetation unit were measured and variations in these factors used to account for the distribution of the vegetation units within the valley. The valley floor is broad and flat and situated below a series of spring lines. The majority of the area has a substrate composed of peat and the importance of water table height, ground water chemistry and tne degree of decomposition of this peat are emphasised as being the major environmental factors controlling the distribution of the main vegetation units of the valley floor. In a small area of the valley floor, which has willow car growing upon it, the substrate is composed of mineral soil. Here the chemistry of the water supplying the area and the nature of the substrate are important environmental factors. The valley sides are steep and composed of a sandy substrate and the dry nature and base-poor conditions of this habitat, seem to, in all probability account for the type of vegetation found growing on these sites. In a study of the population dynamics of the major tree species of the vegetation units, evidence was obtained showing that a change in tree species composition is taking place. Alder and Birch are the main dominants of the valley floor at present, but the evidence presented here suggests that these species are not regenerating and that ash is moving in to become the dominant tree of the valley floor. However, the small area of willow carr presently found in the valley will probably remain with such a species composition, but with a change in the predominant willow species. Further a small area of birch wood with a moss carpet ground layer appears to be losing its tree component and will probably change to a sphagnum bog. The tree composition of the valley sides also appears to be changing. On the fell side birch is being replaced by oak while, on the opposite side of the valley, sycamore is moving in to replace the birch. Reasons for these observed changes are given