This is about the *only* proof I can come up with for **Sperner's theorem** (which see). It's an interesting question whether other, structurally different, proofs exist. In particular, a truly probabilistic proof (not just rephrasing the counting argument below in terms of probability) would be desirable (to me, at least).

### Proof.

Suppose we have some **antichain** F on {1,...,n}. Consider all ordered **permutations** a_{1},a_{n}∈{1,...,n} (that is, all *orderings* of the numbers 1,...,n). There are n! such orderings. Now, the sequence of sets

{a_{1}} ⊆ {a_{1},a_{2}} ⊆ ... ⊆ {a_{1},a_{2},...,a_{n}}

contains at most one

element of F (otherwise F is no antichain).

Suppose f∈F has k elements. Then there are exactly k!(n-k)! orderings that intersect f. So, if c_{k} is the number of elements of F with k elements, by intersections of orderings we have that

∑_{k=0}^{n} c_{k}⋅k!(n-k)! ≤ n!

Now, we're interested in just ∑c

_{k} = |F| (the number of elements of F). But for all k we have that

k!(n-k)! ≥ ⌊n/2⌋!⌈n/2⌉! = m(n)

(this is just the claim that the largest binomial coefficient is the central one). So,

|F|⋅m(n) =
∑_{k=0}^{n} c_{k}⋅m(n) ≤
∑_{k=0}^{n} c_{k}⋅k!(n-k)! ≤
n!,

or, in other words, |F| is not larger than the central binomial coefficient.

**QED.**