The affine type B distance conjecture

Let $w$ be an affine signed permutation in $\widetilde{S}^B_n$. An integer $j\in[n-1]$ is good if $w$ restricts to a bijection from $\pm[j]$ to itself, and it is very good if this restriction is an even-signed permutation. Define

$$\operatorname{bl}^{\widetilde{C}}(w)=1+\#\{\text{good values for }w\},\qquad \operatorname{bl}^{\widetilde{B}}(w)=1+\#\{\text{very good values for }w\}.$$

Here \(w)$and$\operatorname{dis}(w)$denote the quantities defined for affine signed permutations in the paper. **The affine type B conjecture.** If$w\in\widetilde{S}^B_n$, then

$$\$(w)=\frac{1}{2}\operatorname{dis}(w)+\operatorname{bl}^{\widetilde{C}}(w)-\operatorname{bl}^{\widetilde{B}}(w).$$

This conjecture extends the finite-type formula to affine type $\widetilde{B}$. If true, it would imply \(w)-\operatorname{dis}(w)/2\leq n$for all$w\in\widetilde{S}^B_n$; the corresponding boundedness behavior is contrasted with the affine type$\widetilde{D}$ case.