  Let R(n,k) be the number of permutations of $\{1,2,\ldots,n\}$ with k
alternating runs. In this paper, we establish the relationships between R(n,k)
and the central factorial numbers of even indices as well as the number of
signed permutations with a given number of alternating runs and the central
factorial numbers of odd indices. The explicit formulas of the peak and left
peak polynomials for permutations and the derivative polynomials of the tangent
and secant functions are also established.
