Lozenge tilings of a halved hexagon with an array of triangles removed from the boundary

Proctor's work on staircase plane partitions yields an enumeration of lozenge tilings of a halved hexagon on the triangular lattice. Rohatgi recently extended this tiling enumeration to a halved hexagon with a triangle removed from the boundary. In this paper we prove a generalization of the results of Proctor and Rohatgi by enumerating lozenge tilings of a halved hexagon in which an array of adjacent triangles has been removed from the boundary.Comment: 28 pages. Third version: fixed several typo

ESTETIKA

Abstract: In this article some scholars and their perspective on aesthetic are presented, from the beginning of the term to the 20th century (Contemporary aesthetic), althought postmodern aesthetic is not include. The devoloping phases og aesthetic. The Implementation follow the need of aesthetic, wether it is for an evaluation or for other needs. For example, it may be use for the shake of knowledge, to criticize, to deconstruct, and recontruct the work of art. This presentation is hopefully understood and can be implemanted according to the need of aesthetic philosophy. Key word : Filsafat, Estetika dan Sejarah

THE USES OF “ING” FORM IN REFORM MAGAZIN ESECOND EDITION 2003

Mastering structure is a must for language learners, because if the structure of language is not known well it can make errors in comprehending the information. The purpose of this study is to explain the function and to know the frequency of occurrence of “ing” form in Reform magazine second edition 2003. The research design in this study is quantitative research. The population was twenty-five articles that contain ing-form sentences in Reform magazine second edition 20003 and the sample was taken from the whole population. The data collection was obtained through reading the articles and identifying the “ing” forms sentences in the Reform magazine. . The data analysis was conducted by identifying the entire sentences of Reform magazine consist of “ing” forms. Then deciding the function of “ing” forms in each sentence, classifying them and counting the frequency of occurrence and percentage each type in the table. The result of study shows that the uses of “ing” forms in Reform magazine second edition 2003 were dominated by gerund. The total occurrence of gerund which has seven functions is (64.43%) while the total occurrence of participle which has six functions is (35.57%). Concerning with the uses of “ing” forms, suggestion is offered for the teacher to give priority in teaching “ing” form as gerund that is to be the dominant type. For English students they can read more English articles and study how the writer applies the form. The last for the future researcher can conduct the research in the same field with different case of study

A simple proof for the number of tilings of quartered Aztec diamonds

We get four quartered Aztec diamonds by dividing an Aztec diamond region by two zigzag cuts passing its center. W. Jockusch and J. Propp (in an unpublished work) found that the number of tilings of quartered Aztec diamonds is given by simple product formulas. In this paper we present a simple proof for this result