A proof of Lin's conjecture on inversion sequences avoiding patterns of relation triples

A sequence e=e₁e₂⋯e_n of natural numbers is called an inversion sequence if 0≤e_i≤i−1 for all i∈{1,2,…,n}. Recently, Martinez and Savage initiated an investigation of inversion sequences that avoid patterns of relation triples. Let ρ₁, ρ₂ and ρ₃ be among the binary relations {<,>,≤,≥,=,≠,−}. Martinez and Savage defined I_n(ρ₁,ρ₂,ρ₃) as the set of inversion sequences of length n such that there are no indices 1≤i<j<k≤n with e_iρ₁e_j, e_jρ₂e_k and e_iρ₃e_k. In this paper, we will prove a curious identity concerning the ascent statistic over the sets I_n(>,≠,≥) and I_n(≥,≠,>). This confirms a recent conjecture of Zhicong Lin.