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 S̃n. 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 S3× S3, as well as of TITOs that avoid a pair $(p, q)$ with p ∈ S3 ∖ {123, 321} and q ∈ S4. 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.
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BibTeX
@misc{li2026enumerating,
title = {{Enumerating pattern-avoiding translation-invariant total orders}},
author = {Stella Jiahui Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25656},
eprint = {2609.25656},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25656},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Enumerating pattern-avoiding translation-invariant total orders
AU - Stella Jiahui Li
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25656
UR - https://arxiv.org/abs/2609.25656
AB - 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 S̃_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× S_3, as well as of TITOs that avoid a pair $(p, q)$ with p ∈ S_3 ∖ {123, 321} and q ∈ 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.
ER -