Inversion Sequences Avoiding a Triple of Patterns of 3 Letters

An inversion sequence of length $n$ is a sequence of integers $e=e_1\cdots  e_n$ which satisfies for each $i\in[n]=\{1,2,\ldots,n\}$ the inequality $0\le e_i < i$. For a set of patterns $P$, we let $\mathbf{I}_n(P)$ denote the set of inversion sequences of length $n$ that avoid all the patterns from~$P$. We say that two sets of patterns $P$ and $Q$ are I-Wilf-equivalent if $|\mathbf{I}_n(P)|=|\mathbf{I}_n(Q)|$ for every~$n$. In this paper, we show that the number of I-Wilf-equivalence classes among triples of length-3 patterns is $137$, $138$ or~$139$. In particular, to show that this number is exactly $137$, it remains to prove $\{101,102,110\}\stackrel{\mathbf{I}}{\sim}\{021,100,101\}$ and $\{100,110,201\}\stackrel{\mathbf{I}}{\sim}\{100,120,210\}$.