On the Basis Polynomials in the Theory of Permutations with Prescribed   Up-Down Structure
  Let $\pi=(\pi_1,\pi_2,\hdots,\pi_n)$ be permutation of the elements $1,2,\hdots,n. $ Positive integer $k\leq2^{n-1}$ we call index of $\pi,$ if in its binary notation as $n$-digital binary number, the 1's correspond to the ascent points. We study behavior and properties of numbers of permutations of $n$ elements having index $k.$ 
Vladimir Shevelev