321-polygon-avoiding permutations and Chebyshev polynomials

A 321-k-gon-avoiding permutation pi avoids 321 and the following four patterns: k(k+2)(k+3)...(2k-1)1(2k)23...(k+1), k(k+2)(k+3)...(2k-1)(2k)123...(k+1), (k+1)(k+2)(k+3)...(2k-1)1(2k)23...k, (k+1)(k+2)(k+3)...(2k-1)(2k)123...k. The 321-4-gon-avoiding permutations were introduced and studied by Billey and Warrington [BW] as a class of elements of the symmetric group whose Kazhdan-Lusztig, Poincare polynomials, and the singular loci of whose Schubert varieties have fairly simple formulas and descriptions. Stankova and West [SW] gave an exact enumeration in terms of linear recurrences with constant coefficients for the cases k=2,3,4. In this paper, we extend these results by finding an explicit expression for the generating function for the number of 321-k-gon-avoiding permutations on n letters. The generating function is expressed via Chebyshev polynomials of the second kind.Comment: 11 pages, 1 figur

Bulgaria’s Protests: Upholding principles and breaking the political impasse

A casual observer of current events might conclude that Bulgarians are capricious and excitable people who should beware of what they wish for. Last February widespread demonstrations precipitated the resignation of the centre-right government followed by the dissolution of parliament just two months before the end of its term. Having gone through an early general election in mid-May that brought a centre-left government to power, the country has again been gripped by new vigorous political protests. The current campaign builds upon previous waves of discontent, particularly on environmental issues and energy policy where fraud and mismanagement seem to be endemic. However, today’s rallies are primarily driven by insistence on upholding the fundamental principles of democratic rule. Whether the mobilised people will force more accountability, transparency and morality in government and policy-making depends on breaking the precarious balance in the new National Assembly. Together, the two ruling parties, the Socialists (BSP) and the centrist Movement for Rights and Freedoms (MRF) have exactly half the number of deputies and are vulnerable to the increasingly radical antics of an extremely nationalistic formation, Ataka. More importantly, such division represents significant discord in society as a whole

The tourism regions in Bulgaria - concepts and challenges

Bulgaria’s recent experience as tourism country raised the question about the importance of its tourism regions, and the effectiveness of marketing and advertising strategies. Very weak results were achieved about the “new” tourism region planning. The priority based in 2007 NTA (National Tourism Agency) – the regulator which is controlling the tourism policies in Bulgaria, defined the plan for contemporary needs about tourism planning of regions and the usage of that regions for observation, planning and tourism development. It is also about the marketing and advertising activities of the internal and external markets, moreover creating the suitable organizing structures and coordinating of advertising and its tourism policies which is separated in regions is needed. Those policies in tourism must be in regulation and stimulation

Double Object Constructions: A Comparison between their Production by L1 English Speakers, 2L1 English/Spanish Speakers, and L1 Spanish/L2 English Speakers

Este trabajo de fin de grado se basa en el análisis de datos orales espontáneos extraidos de cuatro corpus diferentes: Brown, FerFuLice, Paradis y BELC, todos ellos incluídos en la base de datos CHILDES project (MacWhinney 2000). El objetivo principal de este trabajo es el estudio de la adquisición de la estructura de doble objecto, típica de la lengua inglesa, en distintos tipos de hablantes del inglés, tales como monolingües, bilingües simultáneos que adquieren el español y el inglés desde su nacimiento y bilingües consecutivos cuya primera lengua es el español, con el fin de establecer si las construcciones de doble objeto se adquieren antes que las construcciones de objeto preposicional, si esto ocurre a la misma edad, en el caso de los hablantes nativos, y si existe influencia interlingüística en el caso de los hablantes no nativos de inglés y los bilingües simultáneos. Los resultados obtenidos demuestran que los hablantes cuya lengua materna es el inglés empiezan a producir construcciones de doble objeto a la misma edad y en ambos casos antes de empezar a producir construcciones de objeto preposicional. Así mismo, hay indicios de influencia interlingüística en aquellos hablantes bilingües que tienen como lenguas maternas tanto el inglés como el españolDepartamento de Filología InglesaGrado en Estudios Inglese