A Generalization of Descent Polynomials

2021/09/09 by Angel Raychev, Raychev, Angel
Mathematics · Computer Science · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.2109.04519

Abstract

The notion of a descent polynomial, a function in enumerative combinatorics that counts permutations with specific properties, enjoys a revived recent research interest due to its connection with other important notions in combinatorics, viz. peak polynomials and symmetric functions. We define the function \mathfrakd^m(I,n) as a generalization of the descent polynomial and we prove that for any positive integer m, this function is a polynomial in n for sufficiently large n (similarly to the descent polynomial). We obtain an explicit formula for \mathfrakd^m(I,n) when m is sufficiently large. We look at the coefficients of \mathfrakd^m(I,n) in different falling factorial bases. We prove the positivity of the coefficients and discover a combinatorial interpretation for them. This result is similar to the positivity result of Diaz-Lopez et al. for the descent polynomial.

Related

• Descent polynomials
• Some coefficient sequences related to the descent polynomial
• Restricted inversion polynomials
• Descent polynomials for permutations with bounded drop size
• On the descent polynomial of signed multipermutations