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In [[mathematics]] the Robinson-Schensted-[[Knuth]] correspondence is a [[bijection]] between generalized permutations (as 2 lined arrays) and pairs of [[Young_tableau#Tableauxsemi-standard Young Tableaux]]. [[Richard_P._StanleyRichard Stanley]] uses this term for these tableaux and abbreviates it as SSYT Stanley, Richard P., Enumerative Combinatorics, Volume2. Cambridge University Press. ISBN 0-521-55309-1, 0-521-56069-1. Page 316-380. Donald Knuthuses the term generalized Young Tableaux Knuth, Donald E., Per

mutations, matrices, and generalized Young tableaux. Pacific J. Math. Volume 34,Number 3 (1970), 709-727. Available at http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.pjm/1102971948&page=record.

== Introduction ==

The [[Robinson-Schensted algorithm]] helps establish a [[bijective]] mapping between [[permutations]] and pairs of [[Young_tableauYoung Tableaux]], both havingshape \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 defines by:

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

where,

# i_1 \leq i_2 \leq i_3 \ldots \leq i_m# 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 canobtain [[one-to-one]] mappings from these types of permutations to ordered pair

s, (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 pe

rmutation associated with A, defined as:

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

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d{pmatrix}

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

# 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 matrix A = (a_{ij})_{i,j \geq 1} (and by implication permutation p_A) and the ordered pairs (P,Q), where P and Q have the same shape. In addition integer i occurs exactly a_{i1}+ a_{i2} + \ldots + a_{in} times in Q and the integer j occurs exactly a_{1j} + a_{2j} + \ldots + a_{mj} times in

P.

== Combinatorial properties of RSK correspondence ==

===RSK and permutation matrices===

If A is a [[permutation matrix]] then RSK outputs standard Young Tableaux, P,Q of the same shape \lambda. Conversely, ifP,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 [[Partition (number theory)partition]]s 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: Kunth, Donald E., The Art of Programming, Volume 3/ Sorting and Searching. Addison-Wesley, 1973. Page 54-58

'''Theorem 2''': If the permutation \sigma corresponds to a triple(\lambda,P,Q), then the [[Permutation#Product_and_inverseinverse p

ermutation]], \sigma^{-1}, corresponds to (\lambda,Q,P).

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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 [[Permutation#Product_and_inverseinverse]]):

'''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, \ldo

ts,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 correspondingto (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 thenumber 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===

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.

==References=={{Reflist}}