The median Genocchi numbers, Q-analogues and continued fractions

The goal of this paper is twofold. First, we review the recently developed geometric approach to the combinatorics of the median Genocchi numbers. The Genocchi numbers appear in this context as Euler characteristics of the degenerate flag varieties. Second, we prove that the generating function of the Poincar\' e polynomials of the degenerate flag varieties can be written as a simple continued fraction. As an application we prove that the Poincar\' e polynomials coincide with the $q$-version of the normalized median Genocchi numbers introduced by Han and Zeng.