Combinatorics of stable polynomials and correlation inequalities

This thesis contains five papers divided into two parts. In the first part, Papers I–IV, we study polynomials within the field of combinatorics. Here we study combinatorial properties as well as the zero distribution of the polynomials in question. The second part consists of Paper V, where we study correlating events in randomly oriented graphs. In Paper I we give a new combinatorial interpretation of the stationary distribution of the partially asymmetric exclusion process in terms of colored permutations and decorated alternative trees. We also find a connection between the corresponding multivariate partition functions and the multivariate Eulerian polynomials for r-colored permutations. In Paper II we study a multivariate refinement of P -Eulerian polynomials. We show that this refinement is stable (i.e., non-vanishing whenever the imaginary parts of its variables are all positive) for a large class of labeled posets. In Paper III we use the technique of compatible polynomials to prove that the local h-polynomial of the rth edgewise subdivision of the (n−1)dimensional simplex 2V has only real zeros. We generalize the result and study matrices with interlacing preserving properties. In Paper IV we introduce s-lecture hall partitions for labeled posets. We provide generating functions as well as establish a connection between statistics on wreath products and statistics on lecture hall partitions for posets. In Paper V we prove that the events {s → a} (that there exists a directed path from s to a) and {t → b} are positively correlated in a random tournament for distinct vertices a, s, b, t ∈ Kn. We also discuss the correlation between the same events in two random graphs with random orientation.