The relation ⩽LR on some elements of the affine Weyl group C˜n

Let (W,S) be the affine Weyl group of type C˜n with S its Coxeter generator set. Let Λ‾2n+1 be the set of all partitions λ=(λ1,...,λr) of 2n+1 such that ∑j=12k+1λj is odd for any k∈N with 2k+1⩽r. For any J⊊S, let wJ be the longest element in the parabolic subgroup of W generated by J. We define a map ϕ‾:{wJ|J⊊S}⟶Λ‾2n+1 and study the preorder ⩽LR on the set {wJ|J⊊S} and its relation with the partial order ⩽ on the set {ϕ‾(wJ)|J⊊S}, where iterating star operations and primitive pairs play an important role.