An alternative presentation of the Schensted correspondence

Given a permutation w = w1 w2 ⋯ wn, we define a simple algorithm which yields two words λ(w) = λ1 λ2 ⋯ λn and ϕ(w) = ϕ1 ϕ2 ⋯ ϕn, written on the alphabet [1, n] ∪ {•}; the first row of Schenstedʹs P-symbol wR of w is a reordering of the set of integers in λ(w), while the first row of Schenstedʹs Q-symbol w@ of w consists of the iʹs such that ϕi = •. Iterating the construction to the sequence ϕ(w), ϕ2(w), … gives the successive rows of wR and w@. The function ϕ has the following remarkable properties: (a) if w and w′ belong to the same Knuth class then the same holds for ϕ(w) and ϕ(w′); (b) the function inverse commutes with ϕ: ϕ(tw−1) = (ϕ(w))−1. Classical results of Knuth and Schützenberger follow by induction. We also give a simple algorithm for the correspondence ϕ(w) → w, which supplies an algorithm for the inverse of the Schensted correspondence.