Properties of Our Standard Form for Posets
Let P be a labeled poset with n elements and let C be its child form.
Then the maximal elements of P are those elements which appear in no child list in C.
If P is naturally labeled, then it is easy to see that the jth child list of P can contain only numbers from {1,2,..,j-1}. In particular:
The element labeled '1' will be a minimal element, and so the first child list will be {}.
The element labeled 'n' will be a maximal element, and so will not appear in any child list.
Now suppose further that P has k minimal elements and let S be the standard child form for P.
Then it can be seen that the minimal elements of P must be labeled 1, 2, .., k, and so the first k child lists will each be {}.
Proposition. Let P be a naturally labeled poset with n elements and let S be the standard child form for P. Let L be the set of maximal elements of P. Let P' be the poset formed from P by adjoining an element labeled 'n+1' and dropping edges from it to the maximal elements of P. Then the standard child form S' of P' will consist of the child form list S with the sorted set L appended.
The proof of this proposition appears in Chapter III of Gann's Masters project.