Around the q-binomial-Eulerian polynomials. (May 2019)

• Record Type: Journal Article
• Authors: Lin, Zhicong; Wang, David G.L.; Zeng, Jiang
• Is Part Of: European journal of combinatorics. Volume 78(2019)
• Page Start: 105
• Page End: 120
• Publication Date: 2019-05
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2019.01.010
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: We find a combinatorial interpretation of Shareshian and Wachs' q -binomial-Eulerian polynomials, which leads to an alternative proof of their q - γ -positivity using group actions. Motivated by the sign-balance identity of Désarménien–Foata–Loday for the ( des, inv ) -Eulerian polynomials, we further investigate the sign-balance of the q -binomial-Eulerian polynomials. We show the unimodality of the resulting signed binomial-Eulerian polynomials by exploiting their continued fraction expansion and making use of a new quadratic recursion for the q -binomial-Eulerian polynomials. We finally use the method of continued fractions to derive a new ( p, q ) -extension of the γ -positivity of binomial-Eulerian polynomials which involves crossings and nestings of permutations.