q–counting geodesics in the Fibonacci cube

We study geodesics in the Fibonacci cube Γ n . For any two vertices x , y ∈ V ( Γ n ) , we show that the number of geodesics from x to y factors as a multinomial coefficient times a product of Euler numbers, reflecting a decomposition of the differing coordinates into consecutive blocks. This count is a special case of a natural q –analogue based on the flip–time map of a geodesic. In this general case, the generating function factors into a q –multinomial coefficient and a product of q –Euler polynomials. In the simpler case of the hypercube graph, the statistic yields the classical q –factorial.