Partially ordered sets are ubiquitous in mathematics. In this lecture, we will prove that both (i) the finite Boolean algebra 2^[n] and (ii) L(m,n), the set of all partitions with at most m parts and largest part at most n, both satisfy the Sperner property: they are graded posets for which no antichain is larger than the largest rank level. Along the way we will encounter the poset of Young diagrams, quotient posets, q-binomial coefficients, and a few proofs characteristic of enumerative combinatorics. As time permits, we explore the probabilistic treatment of Young diagrams in the asymptotic representation theory of the symmetric group.
Lecture notes are available here.