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

Abstract

A sequence e=e1e2⋯en of natural numbers is called an inversion sequence if 0≤ei≤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 In(ρ1,ρ2,ρ3) as the set of inversion sequences of length n such that there are no indices 1≤i<j<k≤n with eiρ1ej, ejρ2ek and eiρ3ek. In this paper, we will prove a curious identity concerning the ascent statistic over the sets In(>,≠,≥) and In(≥,≠,>). This confirms a recent conjecture of Zhicong Lin.