Maximizing the Mobius function of a poset and the sum of the Betti numbersof the order complex

This article concerns maximizing the Mobius function for different classes of partially ordered sets and the sum of the Betti numbers for their order complexes. First, we study how using various manipulations on posets can he lp limit the search range for the optimal poset. Then we find the sharp upp er bound for the absolute value of the Mobius function on the class of pose ts of bounded width and classify the posets, which achieve this bound. Next , we consider the topological counterpart of the question. We find the shar p bound for the sum of the Betti numbers for the order complexes of arbitra ry posets, posets of bounded width and ranked posets (with given rank funct ion). We finish with a slight correction of the previous result of G. M. Zi egler.