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In mathematics, the Robinson-Schensted-Knuth correspondence, also referred to as the RSK correspondence or RSK algorithm, is a combinatorial bijection between generalized permutations (as 2 lined arrays) and pairs of semi-standard Young Tableaux (SSYT). It is a generalization of the Robinson-Schensted algorithm.

Introduction
The Robinson-Schensted algorithm, which establishes a bijective mapping between permutations and pairs of Young Tableaux, both having shape $$\lambda$$:

$$\pi \longleftrightarrow (\lambda,P,Q) $$

where $$\pi \in S_n $$ is a permutation of order $$n$$ and $$P,Q$$ are Young Tableau of shape $$\lambda$$.

Generalized permutations
A generalized permutation or two-line array $$w$$ is defined by :

$$ p = \begin{pmatrix}i_1 & i_2 & \ldots & i_m\\j_1 & j_2 & \ldots & j_m\end{pmatrix}$$

where,
 * 1) $$i_1 \leq i_2 \leq i_3 \ldots \leq i_m$$
 * 2) if $$i_r = i_s$$ and $$r \leq s$$ then $$j_r \leq j_s$$

Example:

$$p = \begin{pmatrix}1 & 1 & 1 & 2 & 2 & 3 & 3\\ 1 & 3 & 3 & 2 & 2 & 1 & 2\end{pmatrix}$$

By extending the Robinson-Schensted algorithm to generalized permutations we can obtain one-to-one mappings from these types of permutations to ordered pairs, $$(P,Q)$$, where $$P$$ and $$Q$$ are SSYT of the same shape.

The Robinson-Schensted-Knuth correspondence
The Robinson-Schensted-Knuth (RSK) algorithm works almost exactly like the Robinson-Schensted algorithm, the only difference being that RSK takes a generalized permutation as input. Stanley uses this moniker. The basic operation consists of an insertion operation defined as $$P \longleftarrow k$$ of a positive integer $$k$$ into a SSYT $$P$$.

Let $$A = (a_{ij})_{i,j \geq 1}$$ be a matrix with non-negative elements of dimension $$m \times n$$. Let $$p_A$$ be a generalized permutation associated with $$A$$, defined as:

$$p_A = \begin{pmatrix}i_1 & i_2 & \ldots & i_m\\j_1 & j_2 & \ldots & j_m\end{pmatrix}$$

where in addition to the 2 rules specified in the definition of a generalized permutaion $$p_A$$ needs to satisfy:


 * 1) For each $$(i,j)$$, there must be $$a_{ij}$$ values of $$r$$ for which $$(i_r,j_r) = (i,j)$$.

It is easy to see that there is a bijective mapping from $$A$$ to $$p_A$$.

Example: If

$$p_A = \begin{pmatrix}1 & 1 & 1 & 2 & 2 & 3 & 3\\ 1 & 3 & 3 & 2 & 2 & 1 & 2\end{pmatrix}$$

then

$$A = \begin{bmatrix}1 & 0 & 2 \\ 0 & 2 & 0 \\ 1 & 1 & 0 \end{bmatrix}$$

If we apply the RSK algorithm on the permutation $$p_A$$ we get the following theorem called the Robinson-Schensted-Knuth correspondence theorem.

Theorem 1: There is a one-one correspondence between matrices $$A = (a_{ij})_{i,j \geq 1}$$ (and by implication permutations $$p_A$$) and the ordered pairs $$(P,Q)$$, where $$P$$ and $$Q$$ have the same shape. In addition, each integer $$i$$ occurs exactly $$a_{i1} + a_{i2} + \ldots + a_{in}$$ times in $$Q$$ and each integer $$j$$ occurs exactly $$a_{1j} + a_{2j} + \ldots + a_{mj}$$ times in $$P$$.

RSK and 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 $$\sum_{\lambda\vdash n} (f^\lambda)^2= n!$$ where $$\lambda\vdash n$$ means $$\lambda$$ varies over all partitions of $$n$$ and $$f^\lambda$$ is the number of standard Young tableaux of shape $$\lambda$$.

By examining the structure of the Robinson-Schensted-K algorithm we can prove the following theorem:

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 surprising corollary that links 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 tableax  $$P$$

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 $$\{1,2,3, \ldots,n\}$$,

$$I(n) = n!\sum_{k=0}^{\lfloor n/2 \rfloor} \frac{1}{2^kk!(n-2k)!}$$

Symmetry of RSK
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 $$A$$.

Symmetric Matrices

 * 1) Let $$A$$ be an matrix with non-negative entries, then $$A=A^T$$ if and only if $$P = Q$$ where $$A$$ is mapped to $$(P,Q)$$ by the RSK algorithm.
 * 2) 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 $$\alpha_i \in N$$ and $$\sum \alpha_i < \infty$$. Then the map $$A \longmapsto P$$ establishes a bijection between symmetric matrices with row($$A$$) $$= \alpha$$ and SSYT's of type $$\alpha$$.

Applications of the RSK correspondence
Knuth proved a number of important results that involve symmetric functions that can be derived from the RSK correspondence.

Cauchy's Identity
We have,

$$\prod_{i,j} (1 - x_iy_j)^{-1} = \sum_{\lambda} s_{\lambda}(x)s_{\lambda}(y)$$

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 row row($$A$$) $$= \mu$$ and column($$A$$) $$= \nu$$.