Alperen Özdemir: Recurrence relations and limit theorems

Abstract: Many combinatorial objects are associated with polynomials having rational generating functions, such as the order polynomial of a poset or the Ehrhart polynomial of a polytope. The numerators of these generating functions are important combinatorial polynomials, a classical example being the Eulerian polynomial, which counts permutations by their number of descents. For a tower of such objects, for example permutations of growing length or powers of a polytope, the coefficients of the numerators satisfy certain recurrence relations. We observe that these recurrences define martingales. This yields limit theorems with error bounds for the distribution of the coefficients. Some towers lead to non-normal limits.