over nonincreasing nonnegative f on (0, ∞) of integral unity. We show that the minimum is unique and has the value , as was conjectured by Stanley. The minimizing function f₀, , is given parametrically by f₀(x)=(2/π)(sin θ −θ cos θ), X =f_o(x) + 2 cos θ, 0 θ π (2)

for 0 x 2; and f₀(x) = 0 for x 2. Closely related unpublished results have been obtained by Hammersley. We also find the minimum of H(f) subject to the constraints f(0) a and f⁻¹(0) = inf(x:f(x) = 0) b where a and b are given. Proofs of the results for the case of constraints are complicated and will be given elsewhere. Let λ_n be the shape of the random Young tableau with n unit squares obtained from sampling from the Schensted distribution where P(λ_n=n!/π²(λ_n) (3)

where π(λ_n) is the product of the n hook lengths of λ_n. Consider the stochastic processes λ_n(t)=(1/n^1/2)λ(t n^1/2, n 1, t 0 (4)

where λ_n(t) is the height of the tableau λ_n at a horizontal distance t from the corner. We show λ_n → f₀

in the sense of weak convergence in a certain metric, where f₀ is the deterministic function in (2).

Let l(σ_n) denote the length of the longest increasing subsequence of a random permutation σ_n of 1,2,…,n. Hammersley showed that l(ω_n)/n^1/2 →c in probability, n→∞ (6)

Schensted showed that l(σ_n) has the same distribution as λ_n(0) under the distribution (3) on λ_n. It has long been conjectured (apparently first by Baer and Brock) that c = 2. We show here that c 2 as a by-product of (5).