RSK, the Robinson--Schensted--Knuth correspondence

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#The Robinson–Schensted–Knuth correspondence

The Robinson–Schensted–Knuth correspondence (RSK) is a bijection between matrices with nonnegative integer entries and pairs of semistandard Young tableaux of the same shape. Let $M(\mu,\nu)$ denote the set of matrices with entries in $\setN,$ whose row sums are $\mu$ and whose column sums are $\nu.$ The RSK correspondence provides the following bijections, where $\lambda,$ $\mu$ are integer partitions of $n:$

\[\begin{aligned} M(\mu,\nu) & \qquad \rskArrow \qquad \bigcup_{\lambda \vdash n} \SSYT(\lambda,\nu) \times \SSYT(\lambda,\mu) \\ [d]^n & \qquad \rskArrow \qquad \bigcup_{\lambda \vdash n} \SSYT(\lambda,d) \times \SYT(\lambda) \\ \symS_n & \qquad\rskArrow \qquad \bigcup_{\lambda \vdash n} \SYT(\lambda) \times \SYT(\lambda) \\ \{\sigma \in \symS_n : \sigma^2 = e \} & \qquad \rskArrow \qquad \bigcup_{\lambda \vdash n} \SYT(\lambda). \end{aligned}\]

In particular, the first bijection proves the first Cauchy identity for Schur functions. The binary-matrix variant proves the second Cauchy identity from the same section.

For an excellent overview on the different variants of RSK, see C. Krattenthaler’s survey [Kra06]. We follow his numbering for the ordinary, dual, and Burge variants discussed below. See also [App. A.4.3, Ful97] and [Sag01] for more resources on RSK.

This page first gives the insertion algorithms and the basic matrix correspondence, then records the standard dual and Burge variants, Knuth relations, and the most frequently used symmetry properties. The later sections are a guide to important extensions of RSK rather than complete constructions of every variant.

See the evacuation section for how RSK interacts with evacuation. The standard transpose, inverse, and reversal symmetries are collected in the properties section below. S. Nguyen, J. Vulakh, and D. Woodruff define a generalization of RSK for $d$-complete posets [NVW25]. Their construction gives an elementary combinatorial proof of the hook length formula for $d$-complete posets. R. Singh develops a Robinson–Schensted correspondence for partial permutations using the Steinberg variety of matrix Schubert varieties [Sin20]. The correspondence sends a partial permutation to an admissible signed Young diagram together with two standard Young tableaux.

#Row insertion

In order to describe RSK, we need the notion of row insertion. Given a SSYT $T$ and $k \in \setN,$ the row insertion $T \leftarrow k$ is defined recursively as follows, where $T_1$ is the first row of $T$ and $T_{2\sim}$ denotes the SSYT with the first row of $T$ removed.

If $T = \emptyset,$ then $T \leftarrow k$ is just the tableau $k$ .
If $k$ is not smaller than the largest entry in $T_1,$ then $T \leftarrow k$ is given by appending $k$ at the end of the first row of $T.$
Otherwise, find the leftmost entry in $T_1$ larger than $k,$ and let this be $k'.$ The insertion $T \leftarrow k$ is then given as the tableau with first row $T_1,$ except that $k'$ has been replaced by $k,$ and the remaining rows are given by the insertion $T_{2\sim} \leftarrow k'.$

Given a word $w,$ inserting the letters one by one gives the insertion tableau, $\ins(w).$ This is sometimes denoted $P(w).$

Example

Inserting the entries $w = 34112312$ from left to right gives the following sequence of tableaux, where $\ins(w)$ is the last.

$3$ $3$ $4$ $1$ $4$ $3$ $1$ $1$ $3$ $4$ $1$ $1$ $2$ $3$ $4$ $1$ $1$ $2$ $3$ $3$ $4$ $1$ $1$ $1$ $3$ $2$ $4$ $3$ $1$ $1$ $1$ $2$ $2$ $3$ $3$ $4$

Theorem (Greene’s theorem, [Gre74, Sag01]).

Let $\pi$ be a permutation and let $I_k(\pi)$ denote the length of the longest subword of $\pi$ that can be expressed as a disjoint union of $k$ increasing subsequences of $\pi.$ Similarly, let $D_k(\pi)$ be the length of the longest subword of $\pi$ that can be expressed as a disjoint union of $k$ decreasing subsequences of $\pi.$

Let $\lambda$ be the shape of $P(\pi).$ Then \[I_k(\pi) = \lambda_1+\dotsb + \lambda_k \qquad D_k(\pi) = \lambda'_1+\dotsb + \lambda'_k.\]

K. Menon and A. Singh extend this RSK viewpoint to subsequences with prescribed descent restrictions [MS26]. Their statistics depend only on the recording tableau and are computed using RSK together with Schützenberger evacuation.

Example (An application: Erdős–Szekeres theorem).

The Erdős–Szekeres theorem [ES09] states that any sequence of distinct real numbers of length $(r-1)(s-1)+1$ contains a monotonically increasing subsequence of length $r,$ or a monotonically decreasing subsequence of length $s.$

Suppose we have a sequence $w$ of such numbers. Applying row insertion to this sequence gives a tableau $P$ with real entries. This tableau cannot be contained in an $(r-1)\times (s-1)$-rectangle, so either the first row is longer than $r,$ or the first column is longer than $s.$

Application of Greene’s theorem finishes the proof.

#RSK (variant I)

The RSK correspondence is done in two steps. A matrix $A \in M(\lambda,\mu)$ is first turned into a biword $W$ of length $n$ as follows. For each entry $A_{ij},$ the biword $W$ has $A_{ij}$ columns equal to $\binom{i}{j}.$ The columns are ordered lexicographically, with respect to the top entry first.

Given a pair $(P,Q)$ of SSYT of the same shape, the insertion $(P,Q) \leftarrow \binom{i}{j}$ is defined as the pair $(P',Q'),$ where $P' = P \leftarrow j$ and $Q'$ is obtained from $Q$ by putting $i$ in the box where the last insertion took place to get $P'.$

The pair $(P,Q)$ in the correspondence $W \rskArrow (P,Q)$ is obtained by inserting each column $\binom{i}{j}$ from left to right in $W.$ The resulting $(P,Q)$ is a pair of semistandard Young tableaux whenever $W$ corresponds to a matrix $A.$ We refer to $P = \ins(W)$ as the insertion tableau and $Q=\rec(W)$ as the recording tableau.

Example

The word \[W= \begin{pmatrix} 1& 1& 1& 2& 2& 3& 3& 4 \\ 1& 3& 4& 1& 2& 1& 3& 2 \end{pmatrix}\] is mapped to the pair $(P,Q)$

$1$ $1$ $1$ $2$ $2$ $3$ $3$ $4$ $1$ $1$ $1$ $3$ $2$ $2$ $3$ $4$

Theorem (Greene’s theorem for words, [Gre74, Sag01]).

Let $W$ be a biword with lexicographically sorted columns. Let $I_k(W)$ be the maximum length of a subword of the bottom row of $W$ that can be expressed as a disjoint union of $k$ weakly increasing subwords. Let $D_k(W)$ be the analogous maximum length using $k$ strictly decreasing subwords.

Let $\lambda$ be the shape of $\ins(W).$ Then \[I_k(W) = \lambda_1+\dotsb + \lambda_k \qquad D_k(W) = \lambda'_1+\dotsb + \lambda'_k.\]

#Dual RSK (variant II)

The dual notion of row insertion is called column insertion, traditionally indicated with a right-pointing arrow. We now insert into dual SSYT. A dual SSYT is a filling that is the transpose of an SSYT.

Given a dual SSYT $T$ and $k \in \setN,$ the dual insertion $k \to T$ is defined recursively as follows, where $T_1$ is the first row of $T$ and $T_{2\sim}$ denotes the dual SSYT with the first row of $T$ removed.

If $T = \emptyset,$ then $k \to T$ is just the tableau $k$ .
If $k$ is greater than the largest entry in $T_1,$ then $k \to T$ is given by appending $k$ at the end of the first row of $T.$
Otherwise, find the leftmost entry in $T_1$ larger than or equal to $k,$ and let this be $k'.$ The insertion $k \to T$ is then given as the tableau with first row $T_1,$ except that $k'$ has been replaced by $k,$ and the remaining rows are given by the insertion $k' \to T_{2\sim}.$

This series of operations corresponds to inserting boxes in the columns of $T^t,$ hence the name.

We use the left-to-right convention: dual-inserting the letters of a word $w$ gives the transpose of the ordinary insertion tableau of $\rev(w).$ Equivalently, \[\rev(w) \rskArrow P \iff w \rskDualArrow P^t,\] see [Prop. 2.3.14, But94].

Example

Dual-inserting the entries $w=34112312$ from left to right gives the following sequence of tableaux, where $P$ is the last.

$3$ $3$ $4$ $1$ $4$ $3$ $1$ $4$ $1$ $3$ $1$ $2$ $1$ $4$ $3$ $1$ $2$ $3$ $1$ $4$ $4$ $1$ $2$ $3$ $1$ $4$ $1$ $3$ $1$ $2$ $3$ $1$ $2$ $1$ $4$ $3$

Dual RSK is defined on biwords such that there are no duplicate columns. We denote the dual RSK map as $W \rskDualArrow (P,Q),$ and we obtain a pair of tableaux such that $P$ is dual semistandard and $Q$ is semistandard. Let $B(\mu,\nu)$ denote the set of binary matrices with given row and column sums. Dual RSK provides the following bijections:

\[\begin{aligned} B(\mu,\nu) & \qquad \rskDualArrow \qquad \bigcup_{\lambda \vdash n} \SSYT(\lambda',\nu) \times \SSYT(\lambda,\mu) \\ [d]^n & \qquad \rskDualArrow \qquad \bigcup_{\lambda \vdash n} \SSYT(\lambda',d) \times \SYT(\lambda) \\ \symS_n & \qquad\rskDualArrow \qquad \bigcup_{\lambda \vdash n} \SYT(\lambda') \times \SYT(\lambda) \\ \{\sigma \in \symS_n : \sigma^2 = e \} & \qquad \rskDualArrow \qquad \bigcup_{\lambda \vdash n} \SYT(\lambda). \end{aligned}\]

Example (Dual RSK biword insertion).

The matrix and corresponding word \[A = \begin{pmatrix} 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} \qquad W= \begin{pmatrix} 1& 1& 1& 2& 2& 3& 3& 4 \\ 1& 3& 4& 1& 2& 1& 3& 2 \end{pmatrix}\] is mapped to the pair $(P,Q)$

$1$ $2$ $3$ $1$ $2$ $4$ $1$ $3$ $1$ $1$ $1$ $2$ $2$ $3$ $3$ $4$

Proposition (RSK vs. dual RSK).

Let $\pi \in \symS_n.$ The following conditions are equivalent [Thm. 4.1.1, Lee95]:

$\pi \rskArrow (P,Q)$
$n+1-\pi \rskDualArrow (\evac(P^t),Q^t)$
$\rev(\pi) \rskDualArrow (P^t,\evac(Q^t))$
$\revCompl(\pi) \rskArrow (\evac(P^t),\evac(Q^t))$
$\revCompl(\pi) \rskDualArrow (\evac(P),\evac(Q))$

The following extension is proved in [Prop. 2.3.14, But94]. Let $w \in [d]^n$ be a word. Then \[w \rskArrow P \iff \rev(w) \rskDualArrow P^t.\]

This property generalizes to biwords, see the RSK vs. dual RSK section.

#RSK (variant III)

In the usual RSK we require that the biword $W$ is sorted in lexicographic order. Suppose instead that the columns $\binom{i}{j}$ are sorted first on $i,$ and in case of equality are sorted decreasingly with respect to $j.$ Furthermore, we impose the same condition as in dual RSK that two columns may not be identical. We call such biwords Burge words because this ordering was studied in Burge insertion.

For a broader ribbon-tableau setting, M. A. A. v. Leeuwen constructs spin-preserving Knuth correspondences [Lee05]. These extend the Stanton–White correspondence from colored permutations to matrices with entries in $\setN^r,$ producing pairs of semistandard $r$-ribbon tableaux on a fixed $r$-core, and also give an asymmetric version for matrices with entries in $\{0,1\}^r.$

Each binary matrix $B$ gives rise to a Burge word by recording the row and column coordinates of the ones. That is, if $B_{ij}=1$ then $\binom{i}{j}$ appears as a column in the corresponding biword. For example, \[\begin{pmatrix} 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} \qquad \longleftrightarrow \qquad \begin{pmatrix} 1 & 1 & 2 & 2 & 2 & 3 \\ 4 & 1 & 3 & 2 & 1 & 3 \end{pmatrix}\]

In this setting, we write $W \rskbArrow (P,Q)$ if the biword $W$ is first sorted in the Burge fashion and the entries are inserted using row insertion.

For this map, $W \rskbArrow (P,Q)$ has $P$ and $Q^t$ semistandard tableaux of the same shape, giving another set of bijections. We let $B(\mu,\nu)$ denote the set of binary matrices with row sums $\mu$ and column sums $\nu.$

\[\begin{aligned} B(\mu,\nu) & \qquad \rskbArrow \qquad \bigcup_{\lambda \vdash n} \SSYT(\lambda,\nu) \times \SSYT(\lambda',\mu) \\ [n]^n & \qquad \rskbArrow \qquad \bigcup_{\lambda \vdash n} \SYT(\lambda) \times \SSYT(\lambda') \\ \symS_n & \qquad \rskbArrow \qquad \bigcup_{\lambda \vdash n} \SYT(\lambda) \times \SYT(\lambda') \end{aligned}\]

Example (Burge biword insertion).

The matrix and corresponding word \[A = \begin{pmatrix} 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} \qquad W= \begin{pmatrix} 1& 1& 1& 2& 2& 3& 3& 4 \\ 4& 3& 1& 3& 1& 2& 1& 2 \end{pmatrix}\] is mapped to the pair $(P,Q)$ below.

$1$ $1$ $1$ $2$ $2$ $3$ $3$ $4$ $1$ $2$ $3$ $4$ $1$ $2$ $1$ $3$

Proposition (Burge transpose relation).

Let $B$ be a binary matrix. Then \[B \rskbArrow (P,Q) \iff B^T \rskDualArrow (Q,P).\] Using biwords, this can be phrased as \[\binom{w_1}{w_2} \rskbArrow (P,Q) \iff \binom{w_2}{w_1} \rskDualArrow (Q,P).\] This relation is recorded in [Kra06] and [p. 200, Ful97].

#Knuth relations

Let $A$ be an alphabet. The plactic monoid is the quotient $A^*/\equiv,$ where $\equiv$ is the equivalence class generated by the Knuth relations

\[\begin{aligned} xzy \equiv zxy &\quad (x\leq y \lt z) \\ yxz \equiv yzx &\quad (x\lt y \leq z). \end{aligned}\]

Two words $w$ and $w'$ are Knuth-equivalent if and only if they have the same insertion tableau, $T.$ Moreover, the reading word of $T$ is also Knuth-equivalent to $w,$ and it is the unique word in the equivalence class which is the reading word of a semistandard tableau, see [p. 22, Ful97] or [Cor. 2.3.21, But94].

S. Estupiñán-Salamanca and O. Pechenik give a universal characterization of the shifted plactic monoid [EP24]. This parallels the Lascoux–Schützenberger universal property for the ordinary plactic monoid and places Serrano’s shifted plactic monoid in the same structural framework.

#Dual Knuth relations

Two words $w$ and $w'$ are dual Knuth-equivalent if and only if they have the same recording tableau.

An elementary dual Knuth transformation on a permutation $\sigma_1,\dotsc,\sigma_n$ interchanges $\sigma_i =k$ and $\sigma_j =k+1$ if and only if $k-1$ or $k+2$ occur somewhere between.

For example, $\underline{5}34\underline{6}7182$ and $\underline{6}34\underline{5}7182$ are equivalent, since $4$ appears between them.

Two words are dual Knuth equivalent if and only if one can be transformed to the other via elementary dual Knuth transformations, see [p. 200, Ful97].

#Properties of RSK

The following list records useful properties of RSK. Several of the standard symmetry rules can be found in [App. A.4.3, Ful97] and [A.1.2.11, Sta01]. RSK is also closely connected to charge. The descent identity below uses permutation descents and tableau descents.

For $A \in M(\mu,\nu),$ \[A \quad \rskArrow \quad (P,Q) \qquad \iff \qquad A^{T} \quad \rskArrow \quad (Q,P).\]
For a permutation $\pi \in \symS_n,$ the following statements are equivalent, see [A.1.2.11, Sta01].

\[\begin{aligned} \pi \quad &\rskArrow \quad (P,Q) \\ \pi^{-1} \quad &\rskArrow \quad (Q,P) \\ \rev(\pi) \quad &\rskArrow \quad (P^t, \evac(Q)^t) \end{aligned}\]

For dual RSK, we have the same properties: \[\begin{aligned} \pi \quad &\rskDualArrow \quad (P,Q) \\ \pi^{-1} \quad &\rskDualArrow \quad (Q,P) \\ \rev(\pi) \quad &\rskDualArrow \quad (P^t, \evac(Q)^t) \end{aligned}\]

T. J. Ervin, B. Jackson, J. Lane, K. Lee, S. D. Nguyen, J. O'Donohue, and M. Vaughan characterize the permutations whose reverse has the same Robinson–Schensted recording tableau [EJLL+22]. For such permutations the recording tableau has a symmetric hook shape and satisfies an additional simple tableau condition.
For words, $w \equiv w'$ if and only if $\ins(w)=\ins(w').$
If $w \equiv w',$ then $\charge(w)=\charge(w').$
\[\sum_{\sigma \in \symS_n} t^{\des(\sigma)} = \sum_{\lambda} f^{\lambda} \sum_{Q\in\SYT(\lambda)} t^{\des(Q)},\] since RSK maps descents on permutations to descents in tableaux.
If $\pi \rskArrow (P,Q)$ then $\sign(\pi) = \sign(P)\sign(Q)(-1)^e,$ where $\sign(T)$ is defined as $(-1)^{\inv(\rw(T))}$ and $e$ is the total length of the even-indexed rows in $P,$ see [Rei04].

#RSK and standardization

RSK commutes with standardization, so if $W\rskArrow (P,Q)$ then $W'=\std(W)$ is mapped to $(\std(P),\std(Q)).$

Example (RSK and standardization).

The biword \[W=\begin{pmatrix} 1& 1& 1& 2& 2& 3 \\ 1& 2& 2& 1& 1& 2 \end{pmatrix}\] is mapped to the pair

$1 $ $ 1 $ $ 1 $ $ 2$ $ 2 $ $ 2$ $1 $ $ 1 $ $ 1 $ $ 3$ $ 2 $ $ 2$

and the standardization of the word \[W'=\begin{pmatrix} 1& 2& 3& 4& 5& 6 \\ 1& 4& 5& 2& 3& 6 \end{pmatrix}\] is mapped to the following pair of standard Young tableaux.

$1 $ $ 2 $ $ 3 $ $ 6$ $ 4 $ $ 5$ $1 $ $ 2 $ $ 3 $ $ 6$ $ 4 $ $ 5$

Lemma

For $w \in [d]^n,$ \[\rev(w) \rskDualArrow P \implies \std(w) \rskArrow \std(P^t),\] and \[(\rev \circ w) \rskDualArrow P \implies (\rev \circ \std \circ w) \rskDualArrow \std(P^t)^t.\]

Proof

By [Prop. 2.3.14, But94], \[w \rskArrow P \iff \rev(w) \rskDualArrow P^t.\] This together with the fact that usual RSK commutes with standardization gives the first statement. The second statement follows similarly.

#Skew RSK

B. Sagan and R. Stanley introduce several analogues of the Robinson–Schensted algorithm for skew Young tableaux [SS90]. These skew correspondences give combinatorial proofs of identities involving standard skew tableaux and skew Schur functions, and they retain several of the symmetry features of ordinary RSK.

For a later discussion of injections as generalizations of permutations, see N. Lindzey’s FPSAC 2020 talk [Lin20].

#Statistical properties of RSK

M. Marciniak proves a hydrodynamic limit theorem for the Robinson–Schensted–Knuth algorithm [Mar21]. In a related asymptotic direction, M. Marciniak, {. Ma{\'s}lanka, and P. {\'S}niady prove a Poisson limit theorem for bumping routes in the Robinson–Schensted correspondence [MMS21].

M. Gupta gives an expository account of the longest increasing subsequence problem for random permutations [Gup25]. The survey organizes the RSK correspondence, Young tableaux, hook-length formula, Logan–Shepp and Vershik–Kerov limit shapes, and the Baik–Deift–Johansson fluctuation theorem into a single narrative.

#RSK and pattern avoidance

A. Burcroff and C. Defant use RSK to study strong pattern avoidance, where both a permutation and one of its powers avoid a fixed pattern [BD20].

N. Pahuja considers matrices that map to a fixed shape $\lambda$ under RSK and studies those with the minimal number of inversions, computed from the second row in the biword [Pah26]. The original preprint proposed that these minimal matrices are exactly the symmetric Hankel matrices. The author subsequently found a counterexample to this characterization and is preparing a revised version.

#Rimhook RSK

D. W. Stanton and D. E. White define a bijection [Thm. 3, SW85], \[\begin{pmatrix} H(1)& H(2)& \dotsc & H(\ell) \\ H_1& H_2& \dotsc & H_\ell \\ \end{pmatrix} \rskArrow_{d} (P,Q)\] where $H(1), H(2),\dotsc, H(\ell)$ is a sequence of hook shapes of size $d,$ filled with $1,2,\dotsc,\ell$ and $H_1, H_2, \dotsc, H_\ell$ is the same sequence of hook shapes, but the values are permuted. The output is a pair $(P,Q)$ of $d$-rim-hook tableaux of the same shape. The map $\rskArrow_{d}$ can be decomposed into a $d$-tuple of usual RSK-maps.

As a corollary, $d$-hook tableaux of shape $\lambda$ are in bijection with $d$-tuples of standard Young tableaux with disjoint entry sets, whose shapes are given by the $d$-quotient of $\lambda.$

#Hillman–Grassl correspondence

A $\lambda$-array is a filling of the Young diagram of shape $\lambda$ with nonnegative integers. There are no other conditions. A reverse plane partition of shape $\lambda$ is a $\lambda$-array with weakly increasing rows and columns.

The Hillman–Grassl correspondence, introduced in [HG76], is a bijection from the set of $\lambda$-arrays to the set of reverse plane partitions of shape $\lambda.$ Equivalently, it encodes a reverse plane partition by a multiset of rim hooks in the Young diagram of $\lambda$; the hook lengths explain the resulting product enumeration. Sage has an implementation of the map in its tableau tools [Sag19, com08].

#The Pak–Sulzgruber correspondence

The Pak correspondence is a different bijection, defined by I. Pak [Pak01], from reverse plane partitions to $\lambda$-arrays. The inverse of this map is called the Sulzgruber correspondence.

R. Sulzgruber gives this inverse as a rim-hook insertion algorithm for reverse plane partitions [Sul20]. In [GP19], A. Garver and R. Patrias show that Sulzgruber insertion can be phrased as a diagonal RSK correspondence and that its output is controlled by Greene–Kleitman invariants.

#Signed permutations and RSK

Domino insertion is a type $B/C$ analogue of RSK for signed permutations. T. Lam gives a growth-diagram proof of the semistandard domino-Schensted correspondence, extends it to a nonempty $2$-core, and constructs two dual domino-Schensted correspondences [Lam04]. I. Terada’s Robinson–Schensted-type correspondence for a dual pair on spinors is another representation-theoretic type $B/D$ analogue [Ter93].

#Berele insertion for the symplectic case

The Berele insertion is a symplectic version of Schensted insertion. In [Ber86], it is used to prove the identity \[(a_1+ a_1^{-1} + \dotsb + a_n + a_n^{-1})^m = \sum_{\lambda} Q^\lambda_m(n)\schurSp_\lambda(a_1,\dotsc,a_n)\] where $\schurSp_\lambda(a_1,\dotsc,a_n)$ is a symplectic Schur function and $Q^\lambda_m(n)$ is the number of oscillating sequences of diagrams $\emptyset = \lambda^0,\lambda^1,\dotsc,\lambda^n = \lambda$ with length at most $n.$

M. Kobayashi and T. Matsumura formulate a type $C$ RSK correspondence for King tableaux using Berele insertion [KM26]. The $P$-symbol is a King tableau and the $Q$-symbol is a semistandard oscillating tableau, giving a reformulation of S. Sundaram’s correspondence. They also describe type $C$ Knuth equivalence, derive a duality form of the Cauchy identity, and prove symmetry of the semistandard oscillating-tableau generating functions by Bender–Knuth involutions. M. Kobayashi, T. Matsumura, and S. Sugimoto study the generating function of semistandard oscillating tableaux more directly [KMS26]. They prove fundamental-quasisymmetric positivity, symmetry, Schur positivity, and the saturated Newton polytope property.

I. Nteka gives a randomized $q$-deformation of Berele insertion and uses it to prove Littlewood-type identities for $q$-deformed symplectic Schur functions [Nte18].

For an odd-orthogonal analogue involving vacillating tableaux and $SO(2k+1),$ see [Jag19].

#Multiset RSK

Multiset and set-partition versions of RSK replace ordinary letters in a biword by multisets. T. Halverson and T. Lewandowski construct an RSK insertion for set partitions and diagram algebras [HL05]. L. Colmenarejo, R. Orellana, F. Saliola, A. Schilling, and M. Zabrocki generalize this to two-row arrays of multisets [COSS+20]; one resulting bijection sends words of length $k$ in $[n]$ to a pair consisting of a standard Young tableau of size $n$ and a standard multiset tableau of content $[k].$

For $K$-theoretic analogues, A. Buch, A. Kresch, M. Shimozono, H. Tamvakis, and A. Yong use a common generalization of Robinson–Schensted and Edelman–Greene insertion to study stable Grothendieck polynomials [BKST+07]. H. Thomas and A. Yong develop a $K$-theoretic jeu-de-taquin theory for increasing tableaux [TY09].

T. Guo and S. Poznanović use Hecke insertion and jeu-de-taquin for increasing tableaux to study $01$-fillings of stack polyominoes [GP20]. Their bijections preserve the relevant chain-avoidance data and recover a crossing/nesting symmetry for linked partitions.

A. Wilson extends multiset RSK to two-row arrays of multisets whose elements come from two alphabets [Wil24]. This super multiset RSK gives enumerative decompositions for the mixed multiset partition algebra and for polynomial rings in commuting and anticommuting variables.

#Probabilistic RSK

G. Frieden and F. Schreier-Aigner [AF21] introduced a $q,t$-deformation of the Robinson–Schensted correspondence. The output is a probability distribution on several tableaux, and this proves a squarefree version of the Cauchy identity for Macdonald polynomials. In a later paper, they define a $q,t$-version of RSK from which one can prove the dual Cauchy identity for Macdonald polynomials [FS24].

#Geometric RSK

The geometric RSK correspondence is a birational lift of RSK, where the classical RSK is recovered by tropicalization. Whittaker functions then take the place of Schur functions. See [COSZ14] and the Whittaker-function discussion of geometric RSK.

For a different geometric aspect, see [GR23] where Schubert-geometric type questions (points on a curve) are considered.

#Key polynomials and RSK

O. Azenhas and A. Emami construct an RSK analogue for truncated staircases and nonsymmetric Cauchy kernels [AE15]. For key polynomials and Demazure atoms, see also the RSK-type algorithm of S. Assaf and D. Quijada [AQ19].

D. Zhao extends RSK superinsertion to hook multipartitions and uses it to derive super Frobenius formulae for cyclotomic Hecke algebras [Zha19]. For an expository account of Pak–Postnikov’s super RSK correspondence and a super Cauchy identity, see [Row14].

R. Patrias and P. Pylyavskyy define dual filtered graphs as $K$-theoretic analogues of Fomin’s dual graded graphs [PP14]. Their examples include $K$-theoretic analogues of Young’s lattice and shifted Young’s lattice, with associated insertion algorithms and growth rules.

#Further RSK variants and applications

J. S. Bloom and D. Saracino use H. Thomas and A. Yong’s $K$-infusion operator to build new RSK-type variants for words [BS24]. Their correspondence records longest strictly increasing and decreasing subsequences in every subinterval of the word and proves a conjecture of Guo and Poznanović.
E. Gunawan, J. Pan, H. M. Russell, and B. E. Tenner describe the RSK tableaux of Boolean permutations directly from a canonical reduced word [GPRT24]. They also characterize and enumerate the tableaux that arise in this way.
M. Cofie, O. Fugikawa, E. Gunawan, M. Stewart, and D. Zeng relate RSK recording tableaux to box-ball systems [CFGS+22]. For a permutation used as a box-ball state, its Robinson–Schensted recording tableau determines the dynamics.
B. Drucker, E. Garcia, E. Gunawan, A. Rumbolt, and R. Silver compare RSK insertion tableaux with soliton decompositions of box-ball systems [DGGR+23]. They prove equality in several natural cases, including when the soliton decomposition is a standard tableau.
T. J. Ervin, B. Jackson, J. Lane, K. Lee, S. D. Nguyen, J. O'Donohue, and M. Vaughan characterize permutations whose reverse has the same RSK recording tableau [EJLL+21]. They prove the resulting enumeration, which vanishes in even size and has a closed binomial form in odd size.
N. Hage gives a super RSK correspondence for super tableaux over a signed alphabet [Hag22]. The correspondence has a symmetry property proved through a matrix-ball construction and implies a super Littlewood–Richardson rule for super Schur functions.
B. Brubaker, G. Frieden, P. Pylyavskyy, and T. Scrimshaw develop geometric RSK from the viewpoint of geometric crystals and invariant theory [BFPS25]. They describe generators for fields and rings of invariants under the two commuting geometric crystal actions on matrices.
J. Pappe, D. Paul, and A. Schilling study the Burge correspondence through crystal graphs [PPS23]. They characterize hook-shaped simple graphs by peak and valley conditions on Burge arrays, and construct a crystal structure on simple graphs of hook shape.
E. Stern gives a spectral realization of RSK through the degenerate affine Hecke algebra [Ste26]. The Jucys–Murphy spectrum and rectification recover the insertion and recording tableaux of a permutation.
A set-partition version of RSK involves vacillating tableaux: \[B(2k) = \sum_{\lambda} m^{\lambda}_k \cdot m^{\lambda}_k\] where $m^{\lambda}_k$ is the number of vacillating tableaux of shape $\lambda,$ of length $k$ and $B(2k)$ is a Bell number (number of set partitions of $[2k]$), see [HL05]. For a generalization of this, see [BHPY+22].

This suggests a natural question for run-sorted permutations, see [AN21]. The number of run-sorted permutations of size $2k+1$ is given by $B(2k),$ so one can ask whether some RSK-type map sends such permutations to pairs of vacillating tableaux.
RSK has a close connection with promotion and the shadow construction by X. Viennot, see [PS25].
P. L. Guo relates Richardson tableaux to RSK insertion tableaux of noncrossing partial matchings [Guo25]. This gives a bijection between Richardson tableaux of size $n$ and Motzkin paths with $n$ steps, and connects their $q$-enumeration to $q$-Catalan numbers.
M. Parvathi, A. Tamilselvi, and D. Hepsi construct Robinson–Schensted correspondences for elements of the groups $G_r=\setZ_{p^r}\rtimes \setZ_{p^r}^\ast$ and $SG_r=\setZ_{p^{r-1}}\rtimes \setZ_{p^r}^\ast$ [PTH25]. Their correspondence uses $p$-Young tableaux and primitive idempotents in the group algebras, and leads to a lacunary Cauchy identity.
S. N. Karp and M. E. Precup introduce Richardson tableaux as a subset of standard tableaux motivated by Springer fibers and totally nonnegative geometry [KP25]. They characterize these tableaux using evacuation and pairs of reading words, prove that the corresponding Springer-fiber component is a Richardson variety, and enumerate Richardson tableaux by Motzkin numbers.

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