Enumerating pattern-avoiding translation-invariant total orders

Abstract: Let $n$ be a positive integer. A translation-invariant total order (TITO) with period $n$ is a total order of the integers that is invariant under translations by multiples of $n$. These structures arise naturally in the study of Coxeter groups. In particular, real $n$-TITOs are in bijection with biclosed sets of positive roots of the affine symmetric group $\widetilde S_n$. Barkley and Defant recently introduced pattern avoidance for TITOs and used it to define the affine Tamari lattice. The enumeration of TITOs avoiding a single pattern of length $3$ is due to Crites and Barkley--Defant. We extend this work to TITOs avoiding two patterns. Our main results include a complete enumeration of TITOs that avoid a pair of patterns in $S_3\times S_3$, as well as of TITOs that avoid a pair $(p, q)$ with $p \in S_3 \setminus {123, 321}$ and $q \in S_4$. Furthermore, we provide an explicit construction of the inverse of the bijection between $312$-avoiding TITOs and noncrossing arc diagrams, thereby extending the combinatorial framework introduced by Barkley.