Statistics on pattern-avoiding permutations

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.Includes bibliographical references (p. 111-116).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.This thesis concerns the enumeration of pattern-avoiding permutations with respect to certain statistics. Our first result is that the joint distribution of the pair of statistics 'number of fixed points' and 'number of excedances' is the same in 321-avoiding as in 132-avoiding permutations. This generalizes a recent result of Robertson, Saracino and Zeilberger, for which we also give another, more direct proof. The key ideas are to introduce a new class of statistics on Dyck paths, based on what we call a tunnel, and to use a new technique involving diagonals of non-rational generating functions. Next we present a new statistic-preserving family of bijections from the set of Dyck paths to itself. They map statistics that appear in the study of pattern-avoiding permutations into classical statistics on Dyck paths, whose distribution is easy to obtain. In particular, this gives a simple bijective proof of the equidistribution of fixed points in the above two sets of restricted permutations.(cont.) Then we introduce a bijection between 321- and 132-avoiding permutations that preserves the number of fixed points and the number of excedances. A part of our bijection is based on the Robinson-Schensted-Knuth correspondence. We also show that our bijection preserves additional parameters. Next, motivated by these results, we study the distribution of fixed points and excedances in permutations avoiding subsets of patterns of length 3. We solve all the cases of simultaneous avoidance of more than one pattern, giving generating functions which enumerate them. Some cases are generalized to patterns of arbitrary length. For avoidance of one single pattern we give partial results. We also describe the distribution of these statistics in involutions avoiding any subset of patterns of length 3. The main technique consists in using bijections between pattern-avoiding permutations and certain kinds of Dyck paths, in such a way that the statistics in permutations that we consider correspond to statistics on Dyck paths which are easier to enumerate. Finally, we study another kind of restricted permutations, counted by the Motzkin numbers. By constructing a bijection into Motzkin paths, we enumerate them with respect to some parameters, including the length of the longest increasing and decreasing subsequences and the number of ascents.by Sergi Elizalde.Ph.D

Ecophysiology and ecological impacts of an Antarctic invader: the chironomid, Eretmoptera murphyi.

Antarctica has entered a period of rapid, and potentially drastic, change. The combined pressures of anthropogenic climate change, which disproportionately affects the polar regions, and an increase in human activity and connectivity in and around the Antarctic, is opening the least invaded continent on the planet to new species. As ice retreats, terrestrial habitats ripe for colonisation by both humans and non native species are increasing, and so must our knowledge of the biology, ecology and impact of invading species. This thesis explores these issues through the model invasive species, the chironomid, Eretmoptera murphyi Schaeffer (Diptera: Chironomidae), which has successfully colonised Signy Island in the maritime Antarctic, following introduction by humans in the 1960s. Through whole organism experiments and field observations, we confirm parthenogenesis and adult emergence throughout summer on Signy. Physiological studies are employed to assess the midge’s potential to establish further south, and/or cope with climate change. Differing responses to temperature are identified in different life stages, which at various points in the life cycle must endure microclimate temperatures from +30 ºC to -20 ºC, on Signy Island. The impact of microhabitat temperature and moisture conditions on development and overwintering survival is examined, with oviposition sites found to be an important factor in determining reproductive success, especially considering a warming climate. The extent of E. murphyi’s distribution on Signy is updated, doubling previous estimates of its range, and finding that it is on the brink of moving into new valley systems. Where it occurs, the midge is capable of increasing soil nitrates by as much as five times the background levels, bringing nitrogen levels up to that seen in association with seal colonies. As the only true insect on the island, and a significant detritivore, E. murphyi has the potential to affect change to local vegetation and is arguably a new keystone species in this nutrient-poor ecosystem. Existing biosecurity measures in place seem unlikely to limit its spread which appears to be tracking footpaths used by researchers on the island. Larval stages are also able to survive several weeks in sea water, suggesting there is little impediment to its eventual colonisation of other islands and the Antarctic Peninsula, where it would likely flourish. This body of work encompasses a range of disciplines from whole organism biology through to ecosystem function, and highlights the impact that a single, and seemingly innocuous invasive species can have on an Antarctic terrestrial ecosystem

From Assessing to Conserving biodiversity. Conceptual and Practical Challenges

This open access book features essays written by philosophers, biologists, ecologists and conservation scientists facing the current biodiversity crisis. Despite increasing communication, accelerating policy and management responses, and notwithstanding improving ecosystem assessment and endangered species knowledge, conserving biodiversity continues to be more a concern than an accomplished task. Why is it so? The overexploitation of natural resources by our species is a frequently recognised factor, while the short-term economic interests of governments and stakeholders typically clash with the burdens that implementing conservation actions imply. But this is not the whole story. This book develops a different perspective on the problem by exploring the conceptual challenges and practical defiance posed by conserving biodiversity, namely: on the one hand, the difficulties in defining what biodiversity is and characterizing that “thing” to which the word ‘biodiversity’ refers to; on the other hand, the reasons why assessing biodiversity and putting in place effective conservation actions is arduous. ; Features essays that are explicitly critical of the species approach to biodiversity Presents bio-philosophical perspectives on the interaction between biodiversity’s units, levels, and scales Serves as an interdisciplinary contribution to the emergent field of biodiversity studie

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Annual Report of the Board of Regents of the Smithsonian Institution, showing the operations; expenditures, and condition of the Institution to July, 1888

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