Robinson–Schensted–Knuth correspondence

Introduction 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, ..., σn 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}. The first standard tableau is the result of successive insertions; the other standard tableau records the successive shapes of the intermediate tableaux during the construction of . The Schensted insertion easily generalizes to the case where σ has repeated entries; in that case the correspondence will produce a semistandard tableau rather than a standard tableau, but will still be a standard tableau. The definition of the RSK correspondence reestablishes symmetry between the P and Q tableaux by producing a semistandard tableau for as well. Two-line arrays The two-line array (or generalized permutation) corresponding to a matrix is defined as : w_A = \begin{pmatrix}i_1 & i_2 & \ldots & i_m\\j_1 & j_2 & \ldots & j_m\end{pmatrix} in which for any pair that indexes an entry of , there are columns equal to \tbinom{i}{j}, and all columns are in lexicographic order, which means that • i_1 \leq i_2 \leq i_3 \cdots \leq i_m, and • if i_r = i_s\, and r \leq s then j_r \leq j_s. Example The two-line array corresponding to :A=\begin{pmatrix}1&0&2\\0&2&0\\1&1&0\end{pmatrix} is : w_A = \begin{pmatrix}1 & 1 & 1 & 2 & 2 & 3 & 3\\ 1 & 3 & 3 & 2 & 2 & 1 & 2\end{pmatrix} Definition of the correspondence By applying the Schensted insertion algorithm to the bottom line of this two-line array, one obtains a pair consisting of a semistandard tableau and a standard tableau , where the latter can be turned into a semistandard tableau by replacing each entry of by the -th entry of the top line of . One thus obtains a bijection from matrices to ordered pairs, of semistandard Young tableaux of the same shape, in which the set of entries of is that of the second line of , and the set of entries of is that of the first line of . The number of entries in is therefore equal to the sum of the entries in column of , and the number of entries in is equal to the sum of the entries in row of . Example In the above example, the result of applying the Schensted insertion to successively insert 1,3,3,2,2,1,2 into an initially empty tableau results in a tableau , and an additional standard tableau recoding the successive shapes, given by :P\quad=\quad\begin{matrix}1&1&2&2\\2&3\\3\end{matrix}, \qquad Q_0\quad=\quad\begin{matrix}1&2&3&7\\4&5\\6\end{matrix}, and after replacing the entries 1,2,3,4,5,6,7 in successively by 1,1,1,2,2,3,3 one obtains the pair of semistandard tableaux :P\quad=\quad\begin{matrix}1&1&2&2\\2&3\\3\end{matrix}, \qquad Q\quad=\quad\begin{matrix}1&1&1&3\\2&2\\3\end{matrix}. Direct definition of the RSK correspondence The above definition uses the Schensted algorithm, which produces a standard recording tableau , and modifies it to take into account the first line of the two-line array and produce a semistandard recording tableau; this makes the relation to the Robinson–Schensted correspondence evident. It is natural however to simplify the construction by modifying the shape recording part of the algorithm to directly take into account the first line of the two-line array; it is in this form that the algorithm for the RSK correspondence is usually described. This simply means that after every Schensted insertion step, the tableau is extended by adding, as entry of the new square, the -th entry of the first line of , where b is the current size of the tableaux. That this always produces a semistandard tableau follows from the property (first observed by Knuth) that for successive insertions with an identical value in the first line of , each successive square added to the shape is in a column strictly to the right of the previous one. Here is a detailed example of this construction of both semistandard tableaux. Corresponding to a matrix :A=\begin{pmatrix} 0&0&0&0&0&0&0\\ 0&0&0&1&0&1&0\\ 0&0&0&1&0&0&0\\ 0&0&0&0&0&0&1\\ 0&0&0&0&1&0&0\\ 0&0&1&1&0&0&0\\ 0&0&0&0&0&0&0\\ 1&0&0&0&0&0&0\\ \end{pmatrix} one has the two-line array w_A=\begin{pmatrix}2 & 2 & 3 & 4 & 5 & 6 & 6 & 8 \\ 4 & 6 & 4 & 7 & 5 & 3 & 4 & 1 \end{pmatrix}. The following table shows the construction of both tableaux for this example == Combinatorial properties of the RSK correspondence ==

The case of permutation matrices If A is a permutation matrix then RSK outputs standard Young Tableaux (SYT), P,Q of the same shape \lambda. Conversely, if P,Q are SYT having the same shape \lambda, then the corresponding matrix A is a permutation matrix. As a result of this property by simply comparing the cardinalities of the two sets on the two sides of the bijective mapping we get the following corollary: Corollary 1: For each n \geq 1 we have \sum_{\lambda\vdash n} (t_\lambda)^2= n! where \lambda\vdash n means \lambda varies over all partitions of n and t_\lambda is the number of standard Young tableaux of shape \lambda. Symmetry Let A be a matrix with non-negative entries. Suppose the RSK algorithm maps A to (P,Q) then the RSK algorithm maps A^T to (Q,P), where A^T is the transpose of A. Theorem 2: If the permutation \sigma corresponds to a triple (\lambda,P,Q), then the inverse permutation, \sigma^{-1}, corresponds to (\lambda,Q,P). This leads to the following relation between the number of involutions on S_n with the number of tableaux that can be formed from S_n (An involution is a permutation that is its own inverse): Corollary 2: The number of tableaux that can be formed from \{1,2,3, \ldots,n\} is equal to the number of involutions on \{1,2,3, \ldots,n\}. Proof: If \pi is an involution corresponding to (P,Q), then \pi = \pi^- corresponds to (Q,P); hence P = Q. Conversely, if \pi is any permutation corresponding to (P,P), then \pi^- also corresponds to (P,P); hence \pi = \pi^-. So there is a one-one correspondence between involutions \pi and tableaux P The number of involutions on \{1,2,3, \ldots,n\} is given by the recurrence: : a(n) = a(n-1)+(n-1)a(n-2) \, Where a(1) = 1,a(2) = 2. By solving this recurrence we can get the number of involutions on \{1,2,3, \ldots,n\}, : I(n) = n!\sum_{k=0}^{\lfloor n/2 \rfloor} \frac{1}{2^kk!(n-2k)!} Symmetric matrices Let A = A^T and let the RSK algorithm map the matrix A to the pair (P,P), where P is an SSYT of shape \alpha. Let \alpha = (\alpha_1,\alpha_2,\ldots) where the \alpha_i are non-negative integers and \sum_i \alpha_i . Then the map A \longmapsto P establishes a bijection between symmetric matrices with \mathrm{row}(A) = \alpha and SSYT's of weight \alpha. == Applications of the RSK correspondence ==

Cauchy's identity The Robinson–Schensted–Knuth correspondence provides a direct bijective proof of the following celebrated identity for symmetric functions: : \prod_{i,j} (1 - x_iy_j)^{-1} = \sum_{\lambda} s_{\lambda}(x)s_{\lambda}(y) where s_{\lambda} are Schur functions. Kostka numbers Fix partitions \mu,\nu \vdash n, then : \sum_{\lambda\vdash n} K_{\lambda \mu} K_{\lambda \nu} = N_{\mu \nu} where K_{\lambda \mu} and K_{\lambda \nu} denote the Kostka numbers and N_{\mu \nu} is the number of matrices A, with non-negative elements, with \mathrm{row}(A) = \mu and \mathrm{column}(A) = \nu. == References ==