Degenerate flag varieties and the median genocchi numbers

We study the G^M_a degenerations F^a _λ of the type A flag varieties F_λ. We describe these degenerations explicitly as subvarieties in the products of Grassmannians. We construct cell decompositions of F^a _λ and show that for complete flags the number of cells is equal to the normalized median Genocchi numbers h_n. This leads to a new combinatorial definition of the numbers h_n. We also compute the Poincaré polynomials of the complete degenerate flag varieties via a natural statistics on the set of Dellac's configurations, similar to the length statistics on the set of permutations. We thus obtain a natural q-version of the normalized median Genocchi numbers.