Robinson–Schensted–Knuth correspondence

In mathematics, the Robinson–Schensted–Knuth correspondence, also referred to as the RSK correspondence or RSK algorithm, is a combinatorial bijection between matrices $A$ with non-negative integer entries and pairs $(P, Q)$ of semistandard Young tableaux of equal shape, whose size equals the sum of the entries of $A$ . More precisely the weight of $P$ is given by the column sums of $A$ , and the weight of $Q$ by its row sums. It is a generalization of the Robinson–Schensted correspondence, in the sense that taking $A$ to be a permutation matrix, the pair $(P, Q)$ will be the pair of standard tableaux associated to the permutation under the Robinson–Schensted correspondence.

The Robinson–Schensted–Knuth correspondence extends many of the remarkable properties of the Robinson–Schensted correspondence, notably its symmetry: transposition of the matrix $A$ results in interchange of the tableaux $P, Q$ .

The Robinson–Schensted correspondence is a bijective mapping between permutations and pairs of standard Young tableaux, both having the same shape. This bijection can be constructed using an algorithm called Schensted insertion, starting with an empty tableau and successively inserting the values σ₁,…,σ_n of the permutation σ at the numbers 1,2,…n; these form the second line when σ is given in two-line notation:

$\sigma ={\begin{pmatrix}1&2&\ldots &n\\\sigma _{1}&\sigma _{2}&\ldots &\sigma _{n}\end{pmatrix}}$ .

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