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

2007/12/30 by Vladimir Shevelev, Shevelev, Vladimir
Engineering · Mathematics · #05A15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Basis (linear algebra) #Binomial (polynomial) #Binomial coefficient #Binomial theorem #Combinatorics #Combinatorics (math.CO) #Computer science #Discrete mathematics #Enumeration #FOS: Mathematics #Geometry #Mathematics #Set (abstract data type) #Statistics #graph theory and CDMA systems #math.CO #msc:05A15

paper · pdf · doi:10.48550/arxiv.0801.0072

Revised argument in Section 13; results unchanged

openalex publication_date 2007/12/30 · arxiv created 2010/09/22 · arxiv updated 2010/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Let π=(π₁,π₂,\hdots,π_n) be permutation of the elements 1,2,\hdots,n. Positive integer k≤2^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.

Citations

• Bernoulli-Euler updown numbers associated with function singularities, their combinatorics and arithmetics

Related

• Permutations with given peak set
• The number of permutations with k inversions
• Some coefficient sequences related to the descent polynomial
• Descent polynomials for permutations with bounded drop size
• Number of permutations with same peak set for signed permutations