The (m,n)(m,n)-rational q,tq, t-Catalan polynomials for m=3m=3 and their q,tq,t-symmetry

We introduce a new statistic, skip, on rational $(3,n)$-Dyck paths and define a marked rank word for each path when $n$ is not a multiple of 3. If a triple of valid statistics (area,skip,dinv) are given, we have an algorithm to construct the marked rank word corresponding to the triple. By considering all valid triples we give an explicit formula for the $(m,n)$-rational $q,t$-Catalan polynomials when $m=3$. Then there is a natural bijection on the triples of statistics (area,skips,dinv) which exchanges the statistics area and dinv while fixing the skip. Thus we prove the $q,t$-symmetry of $(m,n)$-rational $q, t$-Catalan polynomials for $m=3$.Comment: 11 pages, 4 figure

Rational Convolution Roots of Isobaric Polynomials

In this paper, we exhibit two matrix representations of the rational roots of generalized Fibonacci polynomials (GFPs) under convolution product, in terms of determinants and permanents, respectively. The underlying root formulas for GFPs and for weighted isobaric polynomials (WIPs), which appeared in an earlier paper by MacHenry and Tudose, make use of two types of operators. These operators are derived from the generating functions for Stirling numbers of the first kind and second kind. Hence we call them Stirling operators. To construct matrix representations of the roots of GFPs, we use the Stirling operators of the first kind. We give explicit examples to show how the Stirling operators of the second kind appear in the low degree cases for the WIP-roots. As a consequence of the matrix construction, we have matrix representations of multiplicative arithmetic functions under the Dirichlet product into its divisible closure.Comment: 13 page

MULTINATIONAL CORPORATIONS AND EU ECONOMIC INTEGRATION

With the development of multinational corporations and the continuous improvement of their status, their influence on world economy and politics is more extensive and profound. In the process of the European Economic integration, investment, production and trade of multinational companies in Eu-rope deepened the international division of labor and economic contact among European countries. The purpose of this thesis is to research the influence of multinational corporations on EU economic integration and to prove their importance on economic integration and global market. The thesis is based on European Union, the world's largest, highest degree of regional eco-nomic inte-gration organization. Starting from the EU multinational companies, it analyzed the influence of polit-ical and economic behavior on EU economic integration. Case analysis method is used in this thesis to research the connection between multinational corporations and economic integration. By research and analysis, the results show that multinational corporations have a powerful impact in improving economic integration