Birkhoff down-set counts: |J(chain_3)|=4, |J(antichain_3)|=8, |J(V)|=5

Exact enumeration of down-sets J(P) for three canonical 3-element posets, demonstrating Birkhoff's representation in action. (1) Chain-3 (0 < 1 < 2): down-sets are {}, {0}, {0,1}, {0,1,2} — exactly 4 = n+1 for an n-chain. (2) Antichain-3…