Interlacing log-concavity of the derangement polynomials and the Eulerian polynomials

Let D(n,k) be the set of derangements of [n][n] with kk excedances and d(n,k)d(n,k) be the cardinality of D(n,k). We establish a bijection between D(n,k) and the set of labeled lattice paths of length nn with kk horizontal edges. Using this bijection, we give a direct combinatorial proof of the inequalities d(n,k−1)d(m,l+1)