Restricted inversion polynomials

2025/11/07 by Lee, Jeongwon, Lesnevich, Nathan, Precup, Martha
Biochemistry, Genetics and Molecular Biology · Mathematics · #05A05 #05E16 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Genome Rearrangement Algorithms #Limits and Structures in Graph Theory

paper · doi:10.48550/arxiv.2511.05676

openalex publication_date 2025/11/07 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28

Abstract

For a finite subset I of positive integers, the descent polynomial D(I;n) counts the number of permutations in S_n that have descent set I. We generalize descent polynomials by considering permutations with a specific subset S of common inversions called h-inversions, where h = (h(1), h(2), … ) is a weakly increasing sequence of positive integers such that h(i)> i. We prove that this more general count, denoted by I_h(S;n), is also a polynomial. We give three explicit expansions for I_h(S;n), prove the coefficients for two of these expansions are log-concave, and define a graded generalization.

Citations

• Descent polynomials
• The Betti numbers of regular Hessenberg varieties are palindromic
• Exponents for B-stable ideals

Related

• Descent polynomials
• A Generalization of Descent Polynomials
• On the descent polynomial of signed multipermutations
• Some coefficient sequences related to the descent polynomial
• The number of permutations with k inversions