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

TL;DR

This paper investigates the properties of permutations with a fixed ascent pattern, analyzing their basis polynomials and the combinatorial structure related to their binary representations.

Contribution

It introduces a novel approach to study permutations with prescribed ascent points using basis polynomials and binary notation analysis.

Findings

Characterization of permutation indices via binary ascent points

Formulas for counting permutations with specific ascent structures

Insights into the algebraic properties of basis polynomials

Abstract

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 $k.$