The Robinson–Schensted–Knuth bijection, and the theorems of Schensted, Greene, and Erdös–Szekeres

The Robinson–Schensted–Knuth bijection, denoted by RSK, is a bijection between sequences π of real numbers of length n ≥ 1 and ordered pairs 〈P,Q〉 where P = P (π) is a semi–standard tableau with entries π1, . . . , πn, Q = Q(π) is a standard Young tableau with entries one through n, and P and Q both have shape λ = λ(π) where |λ| = n. Schensted’s theorem, which has been generalized by Greene, states that the length λ1 (λ1) of the first row (column) of λ equals the length of the longest nondecreasing (nonincreasing) subsequence of π. The Erdös–Szekeres theorem states that a sequence π of real numbers of length n + 1 contains a weakly monotone subsequence of length n+1. We use Greene’s theorem to generalize the Erdös–Szekeres theorem in the case k ≤ D where k is Greene’s parameter and D = D(π) is the rank of the Durfee square of the partition λ.