Adaptation of TITO enumeration techniques to other affine Coxeter types

Determine whether the enumerative techniques developed for translation-invariant total orders associated with the affine symmetric group, including the bijection with noncrossing arc diagrams and the decomposition into waxing and waning blocks, can be adapted to analogous translation-invariant total orders for other affine Coxeter groups, such as types \(\widetilde{B}_r\), \(\widetilde{C}_r\), and \(\widetilde{D}_r\).

Background

The paper studies pattern avoidance for translation-invariant total orders (TITOs) associated with the affine symmetric group $\widetilde{S}_n$. Its enumeration methods rely in particular on a bijection between $312$-avoiding TITOs and noncrossing arc diagrams, as well as on the structural decomposition of TITOs into waxing and waning blocks.

Other affine Coxeter groups admit analogous order-theoretic objects. The paper specifically mentions type $\widetilde{C}_r$ TITOs, defined for a positive odd integer $N=2r+1$ by the translation condition $i\preceq j$ if and only if $i+N\preceq j+N$, together with the reversal condition $-j\preceq -i$. It remains unresolved whether the paper’s combinatorial and enumerative framework extends to these broader affine settings.