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.
Contents
- Introduction
- Two line arrays
- Example
- Definition of the correspondence
- Direct definition of the RSK correspondence
- The case of permutation matrices
- Symmetry
- Symmetric matrices
- Cauchys identity
- Kostka numbers
- References
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.
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:
The first standard tableau P is the result of successive insertions; the other standard tableau Q records the successive shapes of the intermediate tableaux during the construction of P.
The Schensted insertion easily generalizes to the case where σ has repeated entries; in that case the correspondence will produce a semistandard tableau P rather than a standard tableau, but Q 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 Q as well.
Two-line arrays
The two-line array (or generalized permutation) wA corresponding to a matrix A is defined as
in which for any pair (i,j) that indexes an entry Ai,j of A, there are Ai,j columns equal to
-
i 1 ≤ i 2 ≤ i 3 ⋯ ≤ i m - if
i r = i s and r ≤ s thenj r ≤ j s
Example
The two-line array corresponding to
is
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 P and a standard tableau Q0, where the latter can be turned into a semistandard tableau Q by replacing each entry b of Q0 by the b-th entry of the top line of wA. One thus obtains a bijection from matrices A to ordered pairs, (P,Q) of semistandard Young tableaux of the same shape, in which the set of entries of P is that of the second line of wA, and the set of entries of Q is that of the first line of wA. The number of entries j in P is therefore equal to the sum of the entries in column j of A, and the number of entries i in Q is equal to the sum of the entries in row i of A.
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 P, and an additional standard tableau Q0 recoding the successive shapes, given by
and after replacing the entries 1,2,3,4,5,6,7 in Q0 successively by 1,1,1,2,2,3,3 one obtains the pair of semistandard tableaux
Direct definition of the RSK correspondence
The above definition uses the Schensted algorithm, which produces a standard recording tableau Q0, 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 Q is extended by adding, as entry of the new square, the b-th entry ib of the first line of wA, 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 wA, 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
one has the two-line array
The following table shows the construction of both tableaux for this example
The case of permutation matrices
If
Corollary 1: For each
Symmetry
Let
In particular for the case of permutation matrices, one recovers the symmetry of the Robinson–Schensted correspondence:
Theorem 2: If the permutation
This leads to the following relation between the number of involutions on
Corollary 2: The number of tableaux that can be formed from
Proof: If
The number of involutions on
Where
Symmetric matrices
Let
Cauchy's identity
The Robinson–Schensted–Knuth correspondence provides a direct bijective proof of the following celebrated identity for symmetric functions:
where
Kostka numbers
Fix partitions
where