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On the Basis Polynomials in the Theory of Permutations with Prescribed Up-Down Structure

Let $π=(π_1,π_2,\hdots,π_n)$ be permutation of the elements $1,2,\hdots,n. $ Positive integer $k\leq2^{n-1}$ we call index of $π,$ 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 $k.$