In this work we study geometric realizations of permutations of point sets in the plane under strict non-crossing constraints. Specifically, we consider non-self-intersecting paths over sets of n + 1 points in general position with distinct y -coordinates. Each such path has a signature (a word in {-, +} n ), describing the vertical movement along the path. A signature is always-realizable if it can be realized by a non-self-intersecting path for every such point set. A well-known conjecture in upward planar embedding problems is that every signature is always-realizable. We provide new constructive methods that yield broad families of such signatures, including all signatures of length up to 9. We introduce the concept of forward-convex paths, which serves as a key tool for algorithmic path construction. Our framework allows for systematic composition of signature fragments, enabling general constructions. Our results generalize and unify previous work on upward path embeddings onto arbitrary point sets.

In this note we give a negative answer to a question proposed by Almendra-Hernández and Martínez-Sandoval. Let n ≤ m be positive integers and let X and Y be sets of sizes n and m in Rn-1 such that X U Y is in generic position. There is a natural order on X x Y induced by the distances between the corresponding points. The question is if all possible orders on X x Y can be obtained in this way. We show that the answer is negative when n < m. The case n=m remains open.