Monomer-dimer tatami tilings of square regions

We prove that the number of monomer-dimer tilings of an $n\times n$ square grid, with $m<n$ monomers in which no four tiles meet at any point is $m2^m+(m+1)2^{m+1}$, when $m$ and $n$ have the same parity. In addition, we present a new proof of the result that there are $n2^{n-1}$ such tilings with $n$ monomers, which divides the tilings into $n$ classes of size $2^{n-1}$. The sum of these tilings over all monomer counts has the closed form $2^{n-1}(3n-4)+2$ and, curiously, this is equal to the sum of the squares of all parts in all compositions of $n$. We also describe two algorithms and a Gray code ordering for generating the $n2^{n-1}$ tilings with $n$ monomers, which are both based on our new proof.Comment: Expanded conference proceedings: A. Erickson, M. Schurch, Enumerating tatami mat arrangements of square grids, in: 22nd International Workshop on Combinatorial Al- gorithms (IWOCA), volume 7056 of Lecture Notes in Computer Science (LNCS), Springer Berlin / Heidelberg, 2011, p. 12 pages. More on Tatami tilings at http://alejandroerickson.com/joomla/tatami-blog/collected-resource

Negative Correlation Properties for Matroids

In pursuit of negatively associated measures, this thesis focuses on certain negative correlation properties in matroids. In particular, the results presented contribute to the search for matroids which satisfy $$P(\{X:e,f\in X\}) \leq P(\{X:e\in X\})P(\{X:f\in X\})$$ for certain measures, $P$, on the ground set. Let $\mathcal M$ be a matroid. Let $(y_g:g\in E)$ be a weighting of the ground set and let $${Z = \sum_{X}\left( \prod_{x\in X} y_x\right) }$$ be the polynomial which generates Z-sets, were Z $\in \{$ B,I,S $\}$. For each of these, the sum is over bases, independent sets and spanning sets, respectively. Let $e$ and $f$ be distinct elements of $E$ and let $Z_e$ indicate partial derivative. Then $\mathcal M$ is Z-Rayleigh if $Z_eZ_f-ZZ_{ef}\geq 0$ for every positive evaluation of the $y_g$s. The known elementary results for the B, I and S-Rayleigh properties and two special cases called negative correlation and balance are proved. Furthermore, several new results are discussed. In particular, if a matroid is binary on at most nine elements or paving or rank three, then it is I-Rayleigh if it is B-Rayleigh. Sparse paving matroids are B-Rayleigh. The I-Rayleigh difference for graphs on at most seven vertices is a sum of monomials times squares of polynomials and this same special form holds for all series parallel graphs

Facilidad para hacer negocios en la Ciudad de Sullana

La presente tesis tiene como objetivo general analizar mediante los indicadores y metodología del Doing Business, la facilidad de hacer negocios en la ciudad de Sullana en el año 2016, considerando cinco pilares de investigación: (a) apertura de empresa, (b) permiso de construcción, (c) obtención de electricidad, (d) registro de propiedad, y (e) cumplimiento de contratos. Se realizó entrevistas a los principales actores de la economía local, contrastándolo con lo estipulado en gabinete y mediante el análisis DAF con la ciudad de Lima y las principales ciudades de Colombia y España, constituyéndose la naturaleza del estudio como cuantitativa, y el tipo de investigación descriptiva. De acuerdo a los resultados obtenidos hacer negocios en la ciudad de Sullana en los cuatro primeros pilares mencionados, implica realizar un total de 47 trámites. Asimismo, demanda una inversión total en tiempo de 389 días, y en costo de 86.38% del PBI per cápita, constituyendo un total de 17, 980.85 soles. En cuanto a cumplimiento de contratos, implica una inversión en tiempo y costo de 298 días y 6,500 soles (equivalente al 31.22% del PBI per cápita) respectivamente. De igual manera, del análisis realizado se concluye que las entidades que más impacto implica en la facilidad de hacer negocio en la ciudad de Sullana en los primeros cuatro pilares de investigación son: la municipalidad de Sullana, SUNARP, y la Notaria (esta última por presentar particularmente el costo de inversión más alto, a excepción de ENOSA). En ese sentido, conforme lo descrito en el párrafo anterior se propone como recomendación principal que la municipalidad de Sullana asuma el liderazgo para la mejoría de las regulaciones para la facilidad de hacer negocios en la ciudad de Sullana, convocando y aliándose con las demás entidades involucradas, principalmente con la SUNARP y las Notarías, además de todos los actores involucrados, con el objetivo de plantear una estrategia y adicionarla a su Plan de Desarrollo ConcertadoThe general objective of this thesis is to analyze, through the indicators and methodology of the Doing Business, the ease of doing business in the city of Sullana in 2016, considering five research pillars: (a) business start-up, (b) construction permit, (c) Obtaining electricity, (d) registration of property, and (e) compliance with contracts. For that, interviews were conducted with the main actors of the local economy, contrasting with the stipulated in cabinet and DAF analysis with the city of Lima and the main cities of Colombia and Spain, establishing the nature of the study as quantitative, and The type of descriptive research. According to the results obtained doing business in the city of Sullana in the first four pillars mentioned, involves carrying out a total of 47 formalities. It also requires a total investment in time of 389 days, and cost 86.38% of GDP per capita, constituting a total of 17, 980.85 soles. In terms of contract compliance, it involves an investment in time and cost of 298 days and 6,500 soles (equivalent to 31.22% of GDP per capita), respectively. Likewise, from the analysis carried out, it is concluded that the entities that have the greatest impact on the ease of doing business in the city of Sullana in the first four pillars of research are: the municipality of Sullana, SUNARP, and the Notary (particularly the highest investment cost, except for ENOSA). In that sense, as described in the previous paragraph, it is proposed as a main recommendation that the municipality of Sullana assume the leadership for the improvement of regulations for the ease of doing business in the city of Sullana, convening and allying with the other entities involved, Mainly with SUNARP and the Notaries, in addition to all the actors involved, with the aim of presenting a strategy and adding it to its Concerted Development PlanTesi