The systematic study of inversion sequences avoiding triples of relations was initiated by Martinez and Savage. For a triple pρ 1 , ρ 2 , ρ 3 q P tă, ą, ď, ě, ", ‰, ´u3 , they introduced I n pρ 1 , ρ 2 , ρ 3 q as the set of inversion sequences e " e 1 e 2 ¨¨¨e n of length n such that there are no indices 1 ď i ă j ă k ď n with e i ρ 1 e j , e j ρ 2 e k and e i ρ 3 e k . To solve a conjecture of Martinez and Savage, Lin constructed a bijection between I n pě, ‰, ąq and I n pą, ‰, ěq that preserves the distinct entries and further posed a symmetry conjecture of ascents on these two classes of restricted inversion sequences. Concerning Lin's symmetry conjecture, an algebraic proof using the kernel method was recently provided by Andrews and Chern, but a bijective proof still remains mysterious. The goal of this article is to establish bijectively both Lin's symmetry conjecture and the γ-positivity of the ascent polynomial on I n pą, ‰, ąq. The latter result implies that the distribution of ascents on I n pą, ‰, ąq is symmetric and unimodal.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.