In this paper, we consider STATISTICs on permutations of lengthn represented geometrically as bargraphs having the same number of horizontal steps. More precisely, we nd the joint distribution of the descent and up step STATISTICs on the bargraph representations, thereby obtaining a new rened count of permutations of a given length. To do so, we consider the distribution of the parameters on permutations of a more general multiset of which Sn is a subset. In addition to nding an explicit formula for the joint distribution on this multiset, we provide counts for the total number of descents and up steps of all its members, supplying both algebraic and COMBINATORIAL proofs. Finally, we derive explicit expressions for the sign balance of these STATISTICs, from which the comparable results on permutations follow as special cases.

Variance of EWMA STATISTIC is needed when control chart is used. In literatures, a relation is used for computation of this variance which is not exact. We show that this variance is less than the exact value. This paper first derives the exact variance of the EWMA STATISTIC for independent observations. Then through the comparison of this new variance and old variance, the effect of exact variance on control limits of EWMA control chart is examined.

COMBINATORIAL choice models are based on the implicit assumption that decision-makers consider all possible combinations that can be made by the options in a given set. Therefore, these models assumed that the chosen combination is the most preferable combination. However, decision-makers may not consider all possible combinations due to the limited attention. Thus, the chosen combination is not necessarily the best. This paper presents a model that can explain such choice behaviors. After presenting the model, we investigate its revealed preference implications and explain how one can make inferences about individuals’,preferences considering their choices in the new context. Finally, for the model to be testable, we present its characterizing axiom and show that it is equivalent to the model.

Radiology and imaging data of patient is the main and the most important data of the patient. Not only it shows disease of patient, but also it is included of the body geometry such as 2D and 3D. Gathering and saving of radiology images and data for different patients can be useful for future STATISTIC studies. Many questions like, how to, why, in which and etc can be answered by STATISTIC method, if we have had enough information in our hand. In this study we are going to find possibility of fracture in vertebra using STATISTIC method. Radioogle web site was prepared as an search engine to gather information, searching data and result. By this method, the result extracted and show which vertebra are about to harm in case of fracture. The result shows that L1 is the most and L2, L4, L5 and C2 are the next possible to fracture in impact and accident. Other disease of spine and hierarchy in vertebra will be futuristic discussion.

This article is intended to be a survey on some COMBINATORIAL topics in group theory. The bibliography at the end is neither claimed to be exhaustive, nor is it necessarily connected with a reference in the text. I include it as I see it revolves around the concepts which are discussed in the text.