Abstract
The systematic study of inversion sequences avoiding triples of relations was initiated by Martinez and Savage. For a triple (ρ1,ρ2,ρ3)∈{<,>,⩽,⩾,=,≠,−}3, they introduced In(ρ1,ρ2,ρ3) as the set of inversion sequences e=e1e2⋯en of length n such that there are no indices 1⩽i<j<k⩽n with eiρ1ej, ejρ2ek and eiρ3ek. To solve a conjecture of Martinez and Savage, Lin constructed a bijection between In(⩾,≠,>) and In(>,≠,⩾) 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 In(>,≠,>). The latter result implies that the distribution of ascents on In(>,≠,>) is symmetric and unimodal.
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