Let L be a finite distributive lattice. Then I need to prove that the subposet of elements that cover k elements is isomorphic to the subposet of elements that are covered by k elements.
I know the fundamental theorem of finite distributive lattices tells us that L is isomorphic to the poset of order ideals on the join-irreducible elements of L, but I'm not sure how to use that here. The problem appears as number 38 in chapter 3 of Stanley's Enumerative Combinatorics 1, and the solution in the book gives the bijection and says that using FTFDL makes it easy to see why the bijection works, but I really don't have any clue why. The bijection Stanley gives is
phi(t)=sup(u : u not greater than or equal to any x in the join irreducible representation of t)
No clue what's going on here, could anyone help me understand this bijection?
Thes result is false. It is confirmed in the Errata for Stanley's book on his website. Apologies.