Partially Ordered Sets Associated with Permutations

Given a set of permutations ∑,Sn⊇∑, a poset P = P(∑) is chain-permutational with respect to ∑ if it is of minimum cardinality, with the property that its covering relations can be labeled with 1, 2, . . . , n, so that along the maximal chains the labels produce precisely the permutations in ∑, each one exactly once.General properties of chain-permutational posets are discussed, including construction, uniqueness up to isomorphism (in view of which we may use the notation P(∑)), and the fact that P(∑) is a distributive lattice iff ∑ is the collection of linear extensions of a poset.Order-theoretic properties, amongst others, regarding rank cardinalities and shellability are discussed for the special choices of ∑ being the set of involutions in Sn, and ∑ being the pattern restricted permutations in Sn