On the γ-positivity of multiset Eulerian polynomials. (May 2022)

• Record Type: Journal Article
• Authors: Lin, Zhicong; Xu, Chao; Zhao, Tongyuan
• Is Part Of: European journal of combinatorics. Volume 102(2022)
• Publication Date: 2022-05
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2021.103491
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: Let A n ( p ) ( t ) (resp. B n ( p ) ( t ) ) be the descent polynomial on permutations of the multiset { 1 p, 2 p, …, n p } (resp. { 1 p, 2 p, …, n p, n + 1 } ). The γ -positivity of A n ( p ) ( t ) was known but to give a combinatorial interpretation for the corresponding γ -coefficients still remains open. In this paper we manage to find a combinatorial interpretation for the γ -coefficients of A n ( p ) ( t ) via the model of weakly increasing trees; and prove that B n ( p ) ( t ) has bi- γ -positivity expansion B n ( p ) ( t ) = a n ( t ) + t b n ( t ), where b n ( t ) = ( p − 1 ) A n ( p ) ( t ) . The first result, whose proof makes use of Chen's general bijective algorithm for trees and a new decomposition of weakly increasing trees, answers a recent open problem posed by Lin–Ma–Ma–Zhou. The latter result, which we provide both a computational proof and a short combinatorial proof, extends a bi- γ -positivity result due to Ma–Ma–Yeh from p = 2 to general p . Their proof for p = 2 uses Chen's context-free grammar and seems hard to be adopted for general p .