The Pilings of the Mudwater Crossing

The Mudwater is to have a footbridge at last, and a footbridge wants posts driven into the mud. The old carpenter is gone, but the inside of her toolbox lid is covered in notes — one note to a line, each a short rule about how tall the posts stand relative to one another. The carpenters' guild will drive the posts, but only once someone works out how tall each one must be.

Write postHeights(notes).

notes is a list of strings. Each string is exactly three tokens separated by single spaces, and is one of:

• "p = q" — posts p and q stand at the same height
• "p < q" — post p stands strictly lower than post q
• "p : k" — post p stands at exactly height k, where k is a positive integer written in decimal

Post names are non-empty strings of lowercase letters and digits. In = and < notes the two names may be the same name. The posts that exist are exactly the names that appear as posts in notes (the number in a : note is not a post name).

Every post's height is an integer of at least 1. Among all height assignments that satisfy every note, exactly one has the property that no post is taller in it than in any other satisfying assignment; return that one, as a list of [name, height] pairs sorted ascending by name (compared character by character by code point).

If no assignment satisfies every note, return nothing — null in TypeScript, None in Python. If notes is empty, return an empty list.