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\begin{document}

\title{Various representations of the generalized Kostka polynomials}
 
\author{Anatol N. Kirillov}
\address{Division of Mathematics, Graduate School of Science\\
Hokkaido University\\
Sapporo, 060-0810\\
Japan}
\email{kirillov@math.sci.hokudai.ac.jp}
\author{Anne Schilling${}^*$}
\address{Instituut voor Theoretische Fysica\\
Universiteit van Amsterdam\\
Valckenierstraat 65\\
1018 XE Amsterdam\\
The Netherlands}
\email{schillin@wins.uva.nl}
\thanks{${}^*$ Supported by the 
``Stichting Fundamenteel Onderzoek der Materie''.}
\author{Mark Shimozono${}^\dag$}
\address{Department of Mathematics\\
Virginia Tech\\
Blacksburg, VA 24061-0123\\
U.S.A.}
\email{mshimo@math.vt.edu}
\thanks{${}^{\dag}$ Partially supported by NSF grant DMS-9800941.}

\dedicatory{Dedicated to George Andrews on the occasion of his sixtieth
birthday}
\keywords{Generalized Kostka polynomials, charge, path space,
Littlewood--Richardson tableaux, rigged configurations}
\subjclass{Primary 05A19, 05A15; Secondary 81R50, 82B23}
\date{December 1998}

\begin{abstract}
The generalized Kostka polynomials $K_{\la R}(q)$ are labeled by a 
partition $\la$ and a sequence of rectangles $R$. They are $q$-analogues 
of multiplicities of the finite-dimensional irreducible representation 
$W(\la)$ of $\gl_n$ with highest weight $\la$ in the tensor product
$W(R_1)\otimes\cdots\otimes W(R_L)$. We review several representations
of the generalized Kostka polynomials, such as the charge, path space,
quasi-particle and bosonic representation. In addition we describe a 
bijection between Littlewood--Richardson tableaux and rigged 
configurations, and sketch a proof that it preserves the appropriate 
statistics. This proves in particular the equality of the quasi-particle 
and charge representation of the generalized Kostka polynomials.
\end{abstract}
 
\maketitle

\pagestyle{plain}

\section{Introduction}
The Kostka number $K_{\la\mu}$ labeled by two partitions 
$\la=(\la_1,\ldots,\la_n)$ and $\mu=(\mu_1,\ldots,\mu_L)$ is the 
multiplicity of the finite-dimensional irreducible representation 
$W(\la)$ of $\gl_n$ with highest weight $\la$ in the tensor product 
$W(\mu_1)\otimes\cdots\otimes W(\mu_L)$, that is
\begin{equation*}
  K_{\la\mu}=[W(\la):W(\mu_1)\otimes\cdots\otimes W(\mu_L)].
\end{equation*}
This multiplicity is equal to the cardinality of the set of
column-strict Young tableaux of shape $\la$ and content $\mu$.
There is a natural $q$-deformation of the Kostka numbers given
by the Kostka polynomials which are formally defined as the connection
coefficients between the Schur and the Hall--Littlewood 
polynomials~\cite{Mac}. 

There are several explicit expressions for the Kostka polynomials. 
Lascoux and Sch\"utzenberger~\cite{LS} express them as the generating 
function of column-strict Young tableaux with a charge statistic.
They show that the set of column-strict Young tableaux of fixed content
has the structure of a graded poset with the covering relation given
by the cocyclage. In their study of the $XXX$ model using Bethe Ansatz 
techniques, Kirillov and Reshetikhin~\cite{KR} obtained an expression
for the Kostka polynomials in terms of rigged configurations. Rigged
configurations index the solutions of the Bethe Ansatz equations;
they are sequences of partitions obeying certain conditions together
with quantum numbers or riggings labeling the parts of the partitions.
This representation is of interest from the physics perspective as
it reflects the quasi-particle content of the underlying 
statistical mechanical model~\cite{KKMMa,KKMMb}.
A third representation in terms of path spaces was given by Nakayashiki
and Yamada~\cite{NY}, and in a slightly different formulation using
the plactic monoid by Lascoux, Leclerc and Thibon~\cite{LLT95}.
Path spaces first occurred in the corner transfer matrix study of exactly
solvable lattice models (see for example~\cite{Ba}) and are closely
related to the crystal theory of Kashiwara~\cite{Kas}. The statistic
in this case is given by the energy function on paths.
We will refer to these three representations as charge, quasi-particle
and path space representation, respectively.

Recently certain generalizations of the Kostka polynomials were
introduced and studied~\cite{KS,SW,S,S2,S3,ShW}. 
These generalized Kostka polynomials $K_{\la R}(q)$ are labeled by a 
partition $\la$ and a sequence of rectangles $R=(R_1,\ldots,R_L)$, that is, 
each $R_i=(\eta_i^{\mu_i})$ is a partition of rectangular shape. 
They are $q$-analogues of the multiplicity
\begin{equation}\label{mult}
  K_{\la R}(1)=[W(\la):W(R_1)\otimes\cdots\otimes W(R_L)],
\end{equation}
and when all $R_i$ are single rows (in which case $R_i=(\eta_i)$), the
generalized Kostka polynomial reduces to the Kostka polynomial
$K_{\la\eta}(q)$.
The multiplicity $K_{\la R}(1)$ is equal to the cardinality
of the set of Littlewood--Richardson tableaux \cite{FG}.

Akin to the Kostka polynomials, the generalized Kostka polynomials 
have a charge, path space and quasi-particle representation.
We will discuss these representations in Sections~\ref{sec charge}, 
\ref{sec path} and \ref{sec qp}, respectively. A fourth representation
as the sum of $q$-supernomials over the $A_n$ Weyl group 
is given in Section~\ref{sec bose}. $q$-Supernomials extend the 
$q$-binomial coefficients and correspond to the generating function
of unrestricted paths. Because most formulas of the form of this fourth 
representation can be interpreted as (finitizations of) characters or 
branching functions of bosonic algebras, they are referred to as bosonic
representations.

The generalized Kostka polynomials also arise as Poincar\'e 
polynomials of isotypic components of graded $GL(n)$-modules supported 
in the closure of a nilpotent conjugacy class \cite{ShW}. This yields 
in particular another tableau formula for the generalized Kostka polynomials
in terms of catabolizable tableaux \cite[Proposition 19]{S2}.
These connections are however beyond the scope of this paper.

The relations between the different representations of the generalized
Kostka polynomials discussed here have been established in various papers.
The equality between the bosonic and path space representation was shown
in~\cite{SW}, the equality between the path space and the charge
representation was established in~\cite{SW,S3}, and finally
the equivalence of the charge and quasi-particle representation
was recently proven in \cite{KSS}. In \cite{KSS} a bijection between
the set of Littlewood--Richardson tableaux and rigged configurations
is given, which is reviewed in Section \ref{sec bij}. This bijection
preserves the statistics. A sketch of the proof of this property
is given in Section \ref{sec proof}. The equality between 
the quasi-particle and the bosonic representation
can be viewed as a Rogers--Ramanujan-type identity. 
We conclude in Section \ref{sec outlook} with a discussion of some
open problems related to the generalized Kostka polynomials.


\section{Charge representation}
\label{sec charge}

The multiplicity $K_{\la R}(1)$ as defined in \eqref{mult} is equal to
the cardinality of the set of Littlewood--Richardson tableaux.
There are several ways to define LR tableaux.
Here we define the set $\CLR(\la;R)$ where ``C'' indicates a column
labeling. Later we will also need the set of row LR tableaux denoted
by $\RLR(\la;R)$. For a given sequence of rectangles 
$R=(R_1,\ldots,R_L)$ define the standard tableaux $Z_i$ ($1\le i\le L$)
of shape $R_i=(\eta_i^{\mu_i})$ by inserting the numbers
\begin{equation*}
(c-1)\mu_i+\sum_{j=1}^{i-1} |R_i|< k\le c\mu_i+\sum_{j=1}^{i-1}|R_i|
\end{equation*}
into the $c$-th column of $R_i$. For example, for
$R=((2,2),(3,3))$ we have, using the English convention for tableaux,
\begin{equation*}
Z_1=\begin{matrix} 1 & 3 \\ 2 & 4 \end{matrix}\quad\text{and}
\quad Z_2=\begin{matrix} 5 & 7 & 9 \\ 6 & 8 & 10 \end{matrix}.
\end{equation*}
This means that $Z_i$ is a 
standard tableau over the alphabet $B_i=\{|R_1|+\cdots+|R_{i-1}|+1<
\cdots<|R_1|+\cdots+|R_i|\}$.
For a tableau $T$ denote by $T|_B$ the 
restriction of $T$ to the alphabet $B$. The row-reading word 
of a skew tableau $T$ is given by $\word(T)=\cdots w_2 w_1$ where
$w_i$ is the word of the $i$-th row of $T$. Denote by
$P(w)$ the Schensted $P$-tableau of the word $w$ and 
define $P(T):=P(\word(T))$. Finally denote the set of all standard
tableaux of shape $\la$ by $\ST(\la)$.
Then the set $\CLR(\la;R)$ is defined as
\begin{equation*}
  \CLR(\la;R)=\{T\in\ST(\la)|P(T|_{B_i})=Z_i\}.
\end{equation*}

\begin{ex}
Let $\la=(3,2)$ and $R=((1),(2,2))$. Then
\begin{equation*}
T=\begin{matrix} 1&2&4\\3&5&\end{matrix}
\end{equation*}
is in $\CLR(\la;R)$ since $P(T|_{B_1})=1$ and
$P(T|_{B_2})=\begin{matrix}2&4\\3&5\end{matrix}$. On the other hand
$T=\begin{matrix}1&3&5\\2&4\end{matrix}$
is not an LR tableau with respect to $R$ since 
$P(T|_{B_2})=\begin{matrix}2&3&5\\4&&\end{matrix}$.
\end{ex}

It was shown in \cite[Section 6]{SW} and \cite{S} that the set
$\CLR(R)=\cup_{\la}\CLR(\la;R)$ has the structure of a graded poset
with covering relation given by the $R$-cocyclage and grading function
given by the generalized charge, denoted $c_R$.
The generalized Kostka polynomial is the generating function of LR tableaux
with the charge statistic~\cite{SW,S}
\begin{equation}\label{gK}
K_{\la R}(q)=\sum_{T\in\CLR(\la;R)} q^{c_R(T)}.
\end{equation}
This extends the charge representation of the Kostka polynomial
$K_{\la\eta}(q)$ of Lascoux and Sch\"utzenberger~\cite{LS,LS1}.

The charge of an LR tableau can be given explicitly. 
To this end we need to introduce bijections on the
set of LR tableaux which correspond to a permutation of the rectangles
in $R$. Define by $s_p R$ the sequence of rectangles
obtained from $R$ by exchanging $R_p$ and $R_{p+1}$.
Furthermore, denote by $\ev:\ST(\la)\to\ST(\la)$ Sch\"utzenberger's
evacuation involution~\cite{Sc}. This involution restricts to a
bijection $\ev:\CLR(\la;R)\to\CLR(\la;R^{\ev})$ where 
$R^{\ev}=(R_L,\ldots,R_1)$.
\begin{defprop} \label{def bij}
For $1\le p \le L-1$ there are unique
bijections $\sigma_p:\CLR(\la;R)\to\CLR(\la;s_p R)$
satisfying the following properties:
\begin{enumerate}
\item If $p < L-1$ then $\sigma_p$ commutes with restriction to
the initial interval $B_1\cup\cdots\cup B_{L-1}$ where $B_i$ are as in the
definition of $\CLR(\la;R)$.
\item For $p=L-1$ the following diagram commutes:
\begin{equation*}
\begin{CD}
  \CLR(\la;R) @>{\sigma_p}>> \CLR(\la;s_p R) \\
  @V{\ev}VV @VV{\ev}V \\
  \CLR(\la;R^\ev) @>>{\sigma_1}> \CLR(\la;s_1 (R^\ev))
\end{CD}
\end{equation*}
\end{enumerate}
\end{defprop}
\begin{proof} One may reduce to the case $p=L-1$ using 1,
then to the case $p=1$ using 2, and then to the case
$p=1$ and $L=2$ using 1 again.  In this case the
sets of tableaux are all empty or all singletons
by \cite[Prop. 33]{S} and the result holds trivially.
\end{proof}
Then the generalized charge on $\CLR(R)$ is given by~\cite{S,SW}
\begin{equation}\label{charge} 
  \charge_R(T)=\frac{1}{L!}\sum_{\sigma\in S_L}\sum_{i=1}^{L-1}
    (L-i) d_{i,\sigma R}(\sigma T).
\end{equation}
Here $\sigma=\sigma_{p_1}\circ\cdots\circ\sigma_{p_k}$
is an element of the symmetric group,
$d_{i,R}(T)=d_{R_i,R_{i+1}}(\word(T)|_{B_i \cup B_{i+1}})$ where
$B_i$ is the alphabet corresponding to $R_i$ in the definition of 
$\CLR(\la;R)$ and $d_{R_1,R_2}(w)$ is the number of cells of the shape
$P(w)$ that lie in columns strictly to the right of the
$\max(\eta_1,\eta_2)$-th column.

\section{Path space representation}
\label{sec path}

It was shown in~\cite{SW,S3} that the generalized Kostka polynomials
have a path space representation. Denote by $\B_{\la}$ the set of
column-strict Young tableaux of shape $\la$ over the alphabet
$\{1,2,\ldots,n\}$. Let us restrict here to partitions with at most
$n-1$ parts. The partition $\la$ can be identified with
a weight $\La=\La_{\la^t_1}+\La_{\la^t_2}+\cdots$ where $\la^t$ denotes
the transpose of $\la$ and $\La_i$ are the fundamental weights
of $A_{n-1}$. Then the set $\B_{\la}$ is the crystal base of
the irreducible integral highest weight module of highest weight
$\La$ over the quantized universal enveloping algebra 
$U_q(A_{n-1})$~\cite{KN}. The action of the raising and lowering operators 
$e_i$ and $f_i$  on $\B_{\la}$ for $i\in I=\{1,2,\ldots,n\}$ are also given 
in~\cite{KN} which are compatible with the tensor product 
structure. There is an inclusion $U_q(A_{n-1})\subset U_q'(A_{n-1}^{(1)})$
where $U_q'(A_{n-1}^{(1)})$ is the universal enveloping algebra
of the derived algebra ${A'}_{n-1}^{(1)}$~\cite{Kac}. The weight
lattice of ${A'}_{n-1}^{(1)}$ is the $\Z$-span of the fundamental 
weights $\{\La_0,\La_1,\ldots,\La_{n-1}\}$. In ref.~\cite{KKMMNN} the 
actions of $e_i$ and $f_i$ for $i\in J=\{0,1,\ldots,n\}$ were defined, and 
it was shown that for rectangular shapes $\la$ the crystal $\B_{\la}$ 
is perfect.

Write $\phi_i(b)$ (resp. $\epsilon_i(b)$) for the maximum index $m$ such
that $f_i^m(b)\neq 0$ (resp. $e_i^m(b)\neq 0$).
Given crystals $\B_1$ and $\B_2$ define the following crystal
structure on the tensor product $\B_2 \otimes \B_1$ (which differs
from the literature, but is consistent with the Robinson--Schensted--Knuth
correspondence).
For $b_2 \otimes b_1\in \B_2\otimes\B_1$ one defines
\begin{equation*}
\begin{split}
  \phi_i(b_2 \otimes b_1) &= \phi_i(b_2) + 
     \max(0,\phi_i(b_1)-\epsilon_i(b_2)) \\
  \epsilon_i(b_2 \otimes b_1) &= \epsilon_i(b_1) + 
     \max(0,-\phi_i(b_1)+\epsilon_i(b_2)).
\end{split}
\end{equation*}
When $\phi_i(b_2 \otimes b_1) > 0$ (resp. $\epsilon_i(b_2 \otimes b_1) > 0$)
one defines
\begin{align*}
  f_i(b_2 \otimes b_1) &= \begin{cases}
	b_2 \otimes f_i(b_1) & \text{if $\phi_i(b_1)>\epsilon_i(b_2)$} \\
	f_i(b_2) \otimes b_1 & \text{if $\phi_i(b_1)\le\epsilon_i(b_2)$}
	\end{cases} \\
\intertext{and respectively}
  e_i(b_2 \otimes b_1) &= \begin{cases}
	b_2 \otimes e_i(b_1) & \text{if $\phi_i(b_1)\ge\epsilon_i(b_2)$}\\
	e_i(b_2) \otimes b_1 & \text{if $\phi_i(b_1)<\epsilon_i(b_2)$}.
  \end{cases}
\end{align*}

For a partition $\la$ and a sequence of rectangles $R$ define the set
of unrestricted paths
\begin{equation*}
 \PU_{\la R}=\{b_L\otimes\cdots\otimes b_1| 
   \text{$b_i\in\B_{R_i}$ and $\sum_{i=1}^L \content(b_i)=\la$}\}.
\end{equation*}
Say that $b\in \PU_{\la R}$ is classically restricted if
$e_i(b)=0$ for all $i\in I$.
The set of classically restricted paths is defined as
\begin{equation*}
 \PR_{\la R}=\{b\in\PU_{\la R}| 
   \text{$b$ is classically restricted}\}.
\end{equation*}

There exists an energy function on the set of paths.
The definitions here follow \cite{NY}.
Suppose that $\B_1$ and $\B_2$ are perfect crystals of finite-dimensional
$U_q(A_{n-1})$-modules; then $\B_2\otimes \B_1$ is connected.
There is an isomorphism $\B_2 \otimes \B_1 \cong \B_1 \otimes \B_2$.  
This is called the local isomorphism.  Let the image of
$b_2\otimes b_1\in \B_2\otimes \B_1$ under this isomorphism
be denoted $b_1' \otimes b_2'$.  Then there is a unique
(up to global additive constant) map
$H:\B_2 \otimes \B_1 \rightarrow \Z$ such that
\begin{equation*}
  H(e_i(b_2 \otimes b_1)) =  H(b_2\otimes b_1) +
  \begin{cases}
    -1 & \text{if $i=0$, $e_0(b_2\otimes b_1)=e_0 b_2 \otimes b_1$} \\
       & \text{and $e_0(b_1'\otimes b_2')=e_0 b_1' \otimes b_2'$} \\
    1 & \text{if $i=0$, $e_0(b_2\otimes b_1)=b_2 \otimes e_0 b_1$} \\
    & \text{and $e_0(b_1'\otimes b_2')=b_1' \otimes e_0 b_2'$} \\
    0 & \text{otherwise.}
  \end{cases}
\end{equation*}
This map is called the local energy function.

Let $R$ be a sequence of rectangles.
Given $b=b_L \otimes \dots \otimes b_1$ with $b_i\in\B_{R_i}$,
denote by $b_j^{(i+1)}$ the $(i+1)$-th tensor factor in the
image of $b$ under the composition of local isomorphisms that switch
$\B_{R_j}$ with $\B_{R_k}$ as $k$ goes from $j-1$ down to $i+1$.
Then define the energy function
\begin{equation*}
  E(b) = \sum_{1\le i<j\le L} H(b_j^{(i+1)}\otimes b_i).
\end{equation*}

It was shown in~\cite{S3} that the generalized Kostka polynomial can
be expressed as
\begin{equation}\label{K path}
  K_{\la R}(q) = \sum_{b\in \PR_{\la R}} q^{E(b)}.
\end{equation}
This representation extends
the path space representation of the Kostka polynomials of
Nakayashiki and Yamada~\cite{NY}. A slightly different energy function,
more in the spirit of \eqref{charge}, was given in \cite{SW,S3} extending
the representation of \cite{LLT95} in the Kostka case.


\section{Bosonic representation}
\label{sec bose}

Similar to \eqref{K path} of the previous section one may define
the generating function of unrestricted paths as
\begin{equation}\label{S}
  S_{\la R}(q) = \sum_{b\in \PU_{\la R}} q^{E(b)}.
\end{equation}
For $R$ a sequence of single rows or single columns these polynomials 
were studied in \cite{B1,B2,HKKOTY,Ki}. For an arbitrary sequence of
rectangles the polynomials \eqref{S} were introduced in \cite{SW}
where they were called supernomials. The supernomials can be expressed
in terms of the generalized Kostka polynomials and the Kostka
numbers~\cite[Theorem 7.2]{SW}
\begin{equation*}
S_{\la R}(q)=\sum_{\eta} K_{\eta \la}K_{\eta R}(q),
\end{equation*}
where the sum is over all partitions $\eta$ of $|\la|$. The inverse
of this relation yields the bosonic representation of the generalized
Kostka polynomials~\cite[Corollary 7.3]{SW}
\begin{equation*}
 K_{\la R}(q)=\sum_{\tau\in S_n} \epsilon(\tau) S_{(\la_1+\tau_1-1,
  \ldots,\la_n+\tau_n-n)R}(q).
\end{equation*}


\section{Quasi-particle representation}
\label{sec qp}

This section follows~\cite[Section 2.2]{KSS}.

Recall that $R=(R_1,\dots,R_L)$ such that
$R_j$ has $\mu_j$ rows and $\eta_j$ columns for $1\le j \le L$.
For a partition $\la$ denote by $\la^t$ its transpose and set
$R^t=(R_1^t,\dots,R_L^t)$.
A $(\la^t;R^t)$-configuration is a sequence of partitions
$\nu = (\nu^{(1)},\nu^{(2)},\dots)$ with the size constraints
\begin{equation} \label{config def}
  |\nu^{(k)}| = \sum_{j>k} \la^t_j -
  	\sum_{a=1}^L \mu_a \max(\eta_a-k,0)
\end{equation}
for $k\ge 0$ where by convention $\nu^{(0)}$ is the empty partition.
For a partition $\rho$, define $m_n(\rho)$ as the number
of parts equal to $n$ and
$Q_n(\rho)=\rho^t_1+\rho^t_2+\dots+\rho^t_n$,
the size of the first $n$ columns of $\rho$.
Let $\xi^{(k)}(R)$ be the partition whose parts are the heights
of the rectangles in $R$ of width $k$.
The vacancy numbers for the $(\la^t;R^t)$-configuration $\nu$
are the numbers (indexed by $k \ge 1$ and $n\ge 0$) defined by
\begin{equation} \label{vacancy def}
  P^{(k)}_n(\nu) = Q_n(\nu^{(k-1)}) - 2 Q_n(\nu^{(k)}) +
   Q_n(\nu^{(k+1)}) + Q_n(\xi^{(k)}(R)).
\end{equation}
In particular $P^{(k)}_0(\nu) = 0$ for all $k\ge 1$.
The $(\la^t;R^t)$-configuration $\nu$ is
admissible if $P^{(k)}_n(\nu) \ge 0$ for all $k,n \ge 1$, and
the set of admissible $(\la^t;R^t)$-configurations is denoted
by $\C(\la^t;R^t)$. Set
\begin{equation*}
  \cc(\nu)=\sum_{k,n\ge 1} \alpha_n^{(k)}(\alpha_n^{(k)}-\alpha_n^{(k+1)})
\end{equation*}
where $\alpha_n^{(k)}$ is the size of the $n$-th column in
$\nu^{(k)}$. 
\begin{ex}\label{ex config}
Let $\la=(4,3,1)$ and $R=((2),(2,2),(1,1))$. Then 
$\nu=((2),(2,1),(1))$ is a $(\la^t;R^t)$-configuration.
We have $\xi^{(1)}(R)=(2)$ and $\xi^{(2)}(R)=(2,1)$. The configuration
$\nu$ may be represented as
\begin{equation*}
\begin{array}{r|c|c|} \cline{2-3} 1&&\\ \cline{2-3} \end{array}
\qquad
\begin{array}{r|c|c|} \cline{2-3} 0&&\\ \cline{2-3}
      0&&\multicolumn{1}{l}{}\\ \cline{2-2} \end{array}
\qquad
\begin{array}{r|c|} \cline{2-2} 0&\\ \cline{2-2} \end{array}
\end{equation*}
where the vacancy numbers are indicated to the left of each part.
\end{ex}

Define the $q$-binomial as
\begin{equation*}
\qbin{m+n}{m}=\frac{(q)_{m+n}}{(q)_m(q)_n}
\end{equation*}
for $m,n\in\Int$ and zero otherwise where $(q)_m=(1-q)(1-q^2)\cdots
(1-q^m)$.
Then the quasi-particle expression of the generalized Kostka polynomials
can be stated as follows.

\begin{theorem}[Theorem 2.10, \cite{KSS}]\label{theo_qp}
For $\la$ a partition and $R$ a sequence of rectangles
\begin{equation}\label{qp}
  K_{\la R}(q)=\sum_{\nu\in\C(\la^t;R^t)} q^{\cc(\nu)}
   \prod_{k,n\ge 1} \qbin{P_n^{(k)}(\nu)+m_n(\nu^{(k)})}{m_n(\nu^{(k)})}.
\end{equation}
\end{theorem}

Expression \eqref{qp} can be reformulated as the generating function
over rigged configurations. To this end we need
to define certain labelings of the rows of the
partitions in a configuration.
For this purpose one should view a partition as
a multiset of positive integers.
A rigged partition is by definition a finite multiset of
pairs $(n,x)$ where $n$ is a positive integer and
$x$ is a nonnegative integer.  The pairs $(n,x)$ are referred to
as strings; $n$ is referred to as the
length or size of the string and $x$ as the label or
quantum number of the string.  A rigged partition is
said to be a rigging of the partition $\rho$ if
the multiset consisting of the sizes of the strings,
is the partition $\rho$.  So a rigging of $\rho$
is a labeling of the parts of $\rho$ by nonnegative integers,
where one identifies labelings that differ only by
permuting labels among equal-sized parts of $\rho$.

A rigging $J$ of the
$(\la^t;R^t)$-configuration $\nu$ is a sequence of
riggings of the partitions $\nu^{(k)}$ such that
every label $x$ of a part of $\nu^{(k)}$ of size $n$,
satisfies the inequalities
\begin{equation} \label{rigging def}
  0 \le x \le P^{(k)}_n(\nu).
\end{equation}
The pair $(\nu,J)$ is called a rigged configuration.
The set of riggings of admissible $(\la^t;R^t)$-configurations
is denoted by $\RC(\la^t;R^t)$.
Let $(\nu,J)^{(k)}$ be the $k$-th rigged partition
of $(\nu,J)$.  A string $(n,x)\in (\nu,J)^{(k)}$
is said to be singular if $x=P^{(k)}_n(\nu)$, that is,
its label takes on the maximum value.

Observe that the definition
of the set $\RC(\la^t;R^t)$ is completely insensitive to
the order of the rectangles in the sequence $R$.
However the notation involving the sequence $R$
is useful when discussing the bijection 
$\phib_R:\CLR(\la;R)\to\RC(\la^t;R^t)$ between LR tableaux
and rigged configurations as defined in the next section,
since the ordering on $R$ is essential in the definition
of $\CLR(\la;R)$.

The set of rigged configurations is endowed with a natural
statistic $\cc$ \cite[(3.2)]{KS} defined by
\begin{equation*}
  \cc(\nu,J)=\cc(\nu)+\sum_{k,n\ge 1} |J_n^{(k)}|
\end{equation*}
for $(\nu,J)\in\RC(\la^t;R^t)$.
Here $|\rho|$ is the size of the partition $\rho$
and $J_n^{(k)}$ denotes the partition inside the rectangle
of height $m_n(\nu^{(k)})$ and width $P_n^{(k)}(\nu)$ given
by the labels of the parts of $\nu^{(k)}$ of size $n$.
Since the $q$-binomial $\qbins{P+m}{m}$ is the generating
function of partitions with at most $m$ parts each not
exceeding $P$, Theorem \ref{theo_qp} is equivalent to
the following theorem.

\setcounter{theorem}{1}
\renewcommand{\thetheorem}{\arabic{theorem}${}^{\prime}$}
\begin{theorem}[Theorem 2.12, \cite{KSS}]\label{theo_rc}
For $\la$ a partition and $R$ a sequence of rectangles
\begin{equation}\label{rc}
  K_{\la R}(q)=\sum_{(\nu,J)\in\RC(\la^t;R^t)} q^{\cc(\nu,J)}.
\end{equation}
\end{theorem}
\renewcommand{\thetheorem}{\arabic{theorem}}
A detailed proof of this Theorem is given in~\cite{KSS}. 
In the following section we describe the bijection 
$\phi_R:\CLR(\la;R)\to\RC(\la^t;R^t)$ of \cite{KSS}
and sketch the proof in section \ref{sec proof} that this bijection
preserves the statistics, that is $c_R(T)=\cc(\phi_R(T))$.

\section{The bijection between LR tableaux and rigged configurations}
\label{sec bij}

In this section we define the bijection $\phib_R:\CLR(\la;R)\to\RC(\la^t;R^t)$ 
between LR tableaux and rigged configurations. The bijection which
preserves the statistics is
\begin{equation*}
\phi_R=\comp\circ\phib_R
\end{equation*}
where $\comp:\RC(\la;R)\to\RC(\la;R)$ complements the rigging labels.
That is, for $(\nu,J)\in\RC(\la;R)$ a string $(n,x)\in (\nu,J)^{(k)}$
is mapped to $(n,P_n^{(k)}(\nu)-x)$.

The bijection $\phib_R$ is defined recursively based on the following 
two operations on sequences of rectangles $R=(R_1,\ldots,R_L)$:
\begin{enumerate}
\item[I.] Let $\Rht$ be the sequence of rectangles obtained from $R$ by 
splitting off the last column of $R_L$; formally, $\Rht_j = R_j$ for 
$1\le j\le L-1$, $\Rht_L = ((\eta_L-1)^{\mu_L})$ and $\Rht_{L+1}=(1^{\mu_L})$.
\item[II.] If the last rectangle of $R$ is a single column,
let $\Rb$ be given by removing one cell from the column $R_L$;
$\Rb_j=R_j$ for $1\le j \le L-1$ and $\Rb_L = (1^{\mu_L-1})$.
\end{enumerate}

\begin{rem} \label{bij ind}
Given any sequence of rectangles, there is a unique
sequence of transformations of the form $R\rightarrow \Rht$ or
$R\rightarrow \Rb$ resulting in the empty sequence,
where $R\rightarrow \Rht$ is only used when the last rectangle
of $R$ has more than one column.
\end{rem}

For both transformations on sequences of rectangles,
there are natural (injective) maps on the corresponding sets of 
LR tableaux and rigged configurations.
The analogue of transformation I on LR tableaux is the inclusion
\begin{equation*}
  \iht: \CLR(\la;R) \rightarrow \CLR(\la;\Rht).
\end{equation*}
When the last rectangle of $R$ is a single column define
\begin{equation*}
  \CLR(\la^-;\Rb) = \displaystyle\bigcup_{\rho \lessdot \la} \CLR(\rho;\Rb)
\end{equation*}
where $\rho \lessdot\la$ means that $\rho\subset\la$
and $\la/\rho$ is a single cell. 
Define the injective map
\begin{equation*}
\begin{split}
  \CLR(\la;R) &\rightarrow \CLR(\la^-;\Rb) \\
  T &\mapsto T^-
\end{split}
\end{equation*}
where $T^-$ is the LR tableau obtained by removing the
maximum letter from $T$. This corresponds to transform II.

The analogue of transform I for rigged configurations is given by the map
\begin{equation*}
  \jht: \RC(\la^t;R^t) \rightarrow \RC(\la^t;(\Rht)^t)
\end{equation*}
by declaring that $\jht(\nu,J)$ is obtained from $(\nu,J)\in\RC(\la^t;R^t)$
by adding a singular string of length $\mu_L$ to each
of the first $\eta_L - 1$ rigged partitions.
Note that $\jht$ is the identity map if $R_L$ is a single column.
It is shown in \cite[Lemma 3.9]{KSS} that $\jht$ is a well-defined injection
that preserves the vacancy numbers of the underlying configurations.
An example of the map $\jht$ is the first transformation of
Table \ref{tab} in Appendix \ref{sec A}.

Suppose the last rectangle of $R$ is a single column.
Define the set
\begin{equation*}
  \RC(\la^{-t};\Rb^t) = \bigcup_{\rho\lessdot \la} \RC(\rho^t;\Rb^t).
\end{equation*}
The key algorithm on rigged configurations is given by the map
\begin{equation*}
  \jb: \RC(\la^t;R^t) \rightarrow \RC(\la^{-t};\Rb^t),
\end{equation*}
defined as follows.  Let $(\nu,J) \in \RC(\la^t;R^t)$.
Define $\lb^{(0)} = \mu_L$.  By induction select the singular
string in $(\nu,J)^{(k)}$ whose length 
$\lb^{(k)}$ is minimal such that $\lb^{(k-1)} \le \lb^{(k)}$.
Let $\rkb(\nu,J)$ denote the smallest $k$ for which no such
string exists, and set $\lb^{(k)}=\infty$ for $k\ge \rkb(\nu,J)$.
Then $\jb(\nu,J)$ is obtained from $(\nu,J)$ by
shortening each of the selected singular strings by one,
changing their labels so that they remain singular,
and leaving the other strings unchanged.
It is shown in \cite[Proposition 3.12]{KSS} that 
the map $\jb$ is a well-defined injection such that
$\jb(\nu,J) \in \RC(\rho^t;\Rb^t)$ where $\rho$ is obtained from
$\la$ by removing the corner cell in the column of index
$\rkb(\nu,J)$.
\begin{ex}
We continue Example \ref{ex config} and consider the rigged
configuration $(\nu,J)$
\begin{equation*}
\begin{array}{r|c|c|l} \cline{2-3} 1&&&1\\ \cline{2-3} \end{array}
\qquad
\begin{array}{r|c|c|l} \cline{2-3} 0&&&0\\ \cline{2-3}
      0&&\multicolumn{2}{l}{0}\\ \cline{2-2} \end{array}
\qquad
\begin{array}{r|c|l} \cline{2-2} 0&&0\\ \cline{2-2} \end{array}
\end{equation*}
where the vacancy numbers are indicated to the left and the riggings
to the right of each part. By definition $\lb^{(0)}=\mu_2=2$.
Now we need to pick a singular string in $\nu^{(1)}$ of length $\lb^{(1)}$
minimal such that $\lb^{(1)}\ge 2$. The part of length two in $\nu^{(1)}$
is singular; hence $\lb^{(1)}=2$. Next select a singular string in
$\nu^{(2)}$ of length $\lb^{(2)}$ minimal such that $\lb^{(2)}\ge \lb^{(1)}$.
The part of length two in $\nu^{(2)}$ is singular; hence $\lb^{(2)}=2$.
There is no singular string of length greater or equal to two in $\nu^{(3)}$.
This implies $\rkb(\nu,J)=3$. The selected strings are indicated by $*$
in the following diagram
\begin{equation*}
\begin{array}{r|c|c|l} \cline{2-3} 1&&*&1\\ \cline{2-3} \end{array}
\qquad
\begin{array}{r|c|c|l} \cline{2-3} 0&&*&0\\ \cline{2-3}
      0&&\multicolumn{2}{l}{0}\\ \cline{2-2} \end{array}
\qquad
\begin{array}{r|c|l} \cline{2-2} 0&&0\\ \cline{2-2} \end{array}.
\end{equation*}
Now $\jb(\nu,J)$ is obtained by shortening the selected strings
keeping them singular so that $\jb(\nu,J)$ is represented by
\begin{equation*}
\begin{array}{r|c|l} \cline{2-2} 1&&1\\ \cline{2-2} \end{array}
\qquad
\begin{array}{r|c|l} \cline{2-2} 0&&0\\ \cline{2-2}
      0&&0\\ \cline{2-2} \end{array}
\qquad
\begin{array}{r|c|l} \cline{2-2} 0&&0\\ \cline{2-2} \end{array}
\end{equation*}
which is indeed a $(\lab^t;\Rb^t)$-configuration with
$\lab=(4,2,1)$ and $\Rb=((2),(2,2),(1))$.
\end{ex}

The bijection $\phib_R:\CLR(\la;R)\rightarrow\RC(\la^t;R^t)$ is
defined inductively based on Remark \ref{bij ind}.

\begin{defprop} \label{phi def} There is a unique family of bijections
$\phib_R:\CLR(\la;R)\rightarrow \RC(\la^t;R^t)$ indexed by $R$, such that:
\begin{enumerate}
\item If the last rectangle of $R$ is a single column, then
the following diagram commutes:
\begin{equation*}
\begin{CD}
  \CLR(\la;R) @>{-}>> \CLR(\la^-;\Rb) \\
  @V{\phib_R}VV	@VV{\phib_{\Rb}}V \\
  \RC(\la^t;R^t) @>>{\jb}> \RC(\la^{-t};\Rb^t).
\end{CD}
\end{equation*}
\item The following diagram commutes:
\begin{equation*}
\begin{CD}
  \CLR(\la;R) @>{\iht}>> \CLR(\la;\Rht) \\
  @V{\phib_R}VV @VV{\phib_{\Rht}}V \\
  \RC(\la^t;R^t) @>>{\jht}> \RC(\la^t;{\Rht}^t).
\end{CD}
\end{equation*}
\end{enumerate}
\end{defprop}

The proof of this Definition-Proposition is given in \cite{KSS}.
An example illustrating this bijection is given in Appendix \ref{sec A}.


\section{Sketch of the proof of theorem \ref{theo_rc}}
\label{sec proof}

For the proof of Theorem \ref{theo_rc} it remains to show that the bijection
$\phi_R$ preserves the statistics.

\begin{lemma}\label{lem_c}
Let $T\in\CLR(\la;R)$. Then $c_R(T)=\cc(\phi_R(T))$.
\end{lemma}

The proof of this lemma is given in full length in \cite{KSS}. Here we
only sketch the main ideas.

There are further important maps on the sets of LR tableaux and rigged
configurations. The maps which play a central r\^ole in the proof are the
transposition maps on LR tableaux and rigged configurations and a 
statistic preserving embedding on LR tableaux. Let us briefly review
their definitions and some of their properties.

Denote by $\tr:\ST(\la)\to\ST(\la^t)$ the ordinary transposition of
standard tableaux. Analogous to the definition of $\CLR(\la;R)$ let us 
define the set 
\begin{equation*}
\RLR(\la;R)=\{T\in\ST(\la)|P(T|_{B_i})=Z'_i\}
\end{equation*}
where $Z'_i$ is the standard tableau of shape $R_i=(\eta_i^{\mu_i})$ obtained
by inserting the numbers
\begin{equation*}
(r-1)\eta_i+\sum_{j=1}^{i-1}|R_i|<k\le r\eta_i+\sum_{j=1}^{i-1}|R_i|
\end{equation*}
into the $r$-th row of $R_i$.
There is a bijection $\gamma_R:\CLR(\la;R)\to\RLR(\la;R)$ given by 
relabeling as follows. Suppose the letter $j$ occurs in $Z_i$ in cell $s$.
Then, to obtain $\gamma_R(T)$ from $T\in\CLR(\la;R)$, replace the letter $j$
in $T$ by the letter occurring in cell $s$ of $Z'_i$ for all letters $j$.
The transpose map $\tr$ restricts to a bijection $\tr:\CLR(\la;R)
\to\RLR(\la^t;R^t)$. Then the LR-transpose 
\begin{equation*}
\LRtr:\CLR(\la;R)\to\CLR(\la^t;R^t)
\end{equation*}
is defined as $\LRtr:=\tr\circ\gamma_R$.

An analogous RC-transpose bijection exists 
for the set of rigged configurations denoted by 
$\RCtr:\RC(\la^t;R^t)\rightarrow\RC(\la;R)$, which was described
in \cite[Section 9]{KS}. 
Let $(\nu,J)\in\RC(\la^t;R^t)$ and let $\nu$ have
the associated matrix $m$ with entries $m_{ij}$ as in \cite[(9.2)]{KS}
\begin{equation*}
  m_{ij} = \alpha^{(i-1)}_j - \alpha^{(i)}_j
\end{equation*}
for $i,j\ge 1$, where $\alpha^{(i)}_j$ is the size of the
$j$-th column of the partition $\nu^{(i)}$, recalling that
$\nu^{(0)}$ is defined to be the empty partition.
The configuration $\nu^t$ in $(\nu^t,J^t)=\RCtr(\nu,J)$ is defined
by its associated matrix $\mt$ given by
\begin{equation*}
  \mt_{ij} = - m_{ji} +\chi((i,j)\in \la) -
	\sum_{a=1}^L \chi((i,j)\in R_a)
\end{equation*}
for all $i,j \ge 1$. Here $(i,j) \in \la$ means that
the cell $(i,j)$ is in the Ferrers diagram of the partition $\la$
with $i$ specifying the row and $j$ the column, and
$\chi(\mathrm{true})=1$ and $\chi(\mathrm{false})=0$.
Recall that the rigging $J$ is determined by partitions
$J_n^{(k)}$ inside the rectangle of height
$m_n(\nu^{(k)})$ and width $P_n^{(k)}(\nu)$ given by the labels of the parts
of $\nu^{(k)}$ of size $n$. The partition $J_k^{t(n)}$ 
corresponding to $(\nu^t,J^t)=\RCtr(\nu,J)$ is defined as the transpose of 
the complementary partition to $J_n^{(k)}$ in the rectangle of 
height $m_n(\nu^{(k)})$ and width $P_n^{(k)}(\nu)$.

It is shown in \cite[Theorem 7.1]{KSS} that the diagram
\begin{equation} \label{transpose bij}
\begin{CD}
  \CLR(\la;R) @>{\LRtr}>> \CLR(\la^t;R^t) \\
  @V{\phi_R}VV @VV{\phi_{R^t}}V \\
  \RC(\la^t;R^t) @>>{\RCtr}> \RC(\la;R)
\end{CD}
\end{equation}
commutes.

Let $\rows(R)$ be obtained from the sequence of rectangles $R$ by slicing
all the rectangles of $R$ into single rows. In refs. \cite{S,SW} an 
embedding
\begin{equation*}
\theta_R:\CLR(\la;R)\to\CLR(\la;\rows(R))
\end{equation*}
was defined and it was shown that $\theta_R$ preserves the charge $c_R$
of LR tableaux. This embedding stems from an analogous embedding
on column-strict tableaux given by Lascoux and 
Sch\"utzenberger~\cite{L,LS1}.
For rigged configurations, it follows immediately from the definitions
that there is an inclusion $\RC(\la^t;R^t)\subseteq\RC(\la^t;\rows(R)^t)$.
It is shown in \cite[Theorem 8.3]{KSS} that the diagram
\begin{equation} \label{embedding bij}
\begin{CD}
  \CLR(\la;R) @>{\theta_R}>> \CLR(\la;\rows(R)) \\
  @V{\phi_R}VV @VV{\phi_{\rows(R)}}V \\
  \RC(\la^t;R^t) @>>{\mathrm{inclusion}}> \RC(\la^t;\rows(R)^t)
\end{CD}
\end{equation}
commutes. 

Now the proof of Lemma \ref{lem_c} follows directly from 
\eqref{transpose bij} and \eqref{embedding bij}.
Since the embedding $\theta_R$ preserves the statistics one can 
reduce the proof of Lemma \ref{lem_c} to the case that all rectangles in $R$
are single rows using \eqref{embedding bij}. By \eqref{transpose bij} it may
be assumed that $R$ consists of single columns only. Finally applying 
\eqref{embedding bij} again, it is sufficient to establish Lemma~\ref{lem_c}
for $R$ a sequence of single boxes only. In this case the lemma is
verified explicitly~\cite{KSS}.

\section{Outlook}
\label{sec outlook}

Let us take this opportunity to discuss some open problems regarding
the generalized Kostka polynomials.

It was conjectured in~\cite{Ki,KS} that the generalized Kostka polynomials
coincide with special cases of the spin generating functions over
ribbon tableaux of Lascoux, Leclerc and Thibon~\cite{LLT97}. A proof
of this conjecture would add yet another representation of the generalized
Kostka polynomials to the list. The spin generating functions over
ribbon tableaux are defined for an arbitrary sequence of partitions $R$,
not just a sequence of rectangles.

The algebraic structure underlying the generalized Kostka polynomials
is determined by the affine Kac--Moody algebras of type $A$ (compare with 
Section \ref{sec path}). The crystal theory can be formulated very
generally for any affine Kac--Moody algebra. Recently Hatayama et 
al.~\cite{HKOTY} conjectured quasi-particle representations
for all untwisted affine algebras. It would be interesting to
find proofs of these conjectures.

In addition to the sets of unrestricted and classically restricted
paths one can also define the set of level-restricted paths by imposing
additional conditions using the raising operator $e_0$.
Quasi-particle representations for the generating functions of
level-restricted paths have been conjectured (see e.g.~\cite{HKOTY,SW}).
However, proofs only exist in isolated cases (see e.g.~\cite{Be,FOW,SWa}).
A bosonic representation for level-restricted paths was recently proven
in~\cite{SS}.

Finally, it would be interesting to find explicit expressions for the 
supernomial coefficients as introduced in Section~\ref{sec bose}.
So far, explicit formulas for the supernomials are only known
for type $A$ when $R$ is either a sequence of single rows or single
columns.

\appendix
\section{An example of the bijection}
\label{sec A}

\begin{table}
\begin{center}
\begin{equation*}
\begin{array}{|ll|c|c|} \hline &&&\\[-3mm] 
 (\nu,J)^{(1)} & (\nu,J)^{(2)} & R & \rkb(\nu,J) \\[1mm] \hline &&&\\
   \begin{array}{r|c|l} \cline{2-2} 0& &0 \\ \cline{2-2} \end{array}
 & \begin{array}{r|c|l} \cline{2-2} 0& &0 \\ \cline{2-2} \end{array}
 & \begin{array}{|c|} \hline \\ \hline \end{array} \quad
   \begin{array}{|c|c|} \hline & \\ \hline & \\ \hline \end{array}
 &\\[4mm]
 \multicolumn{2}{|c|}{{\Big \downarrow} \jht} &\quad
 {\Big \downarrow} \htt &\\[3mm]
   \begin{array}{r|c|c|l} \cline{2-3} 0&&*&0\\ \cline{2-3}
      0&&\multicolumn{2}{l}{0}\\ \cline{2-2} \end{array}
 & \begin{array}{r|c|l} \cline{2-2} 0& &0 \\ \cline{2-2} \end{array}
 & \begin{array}{|c|} \hline \\ \hline \end{array} \quad
   \begin{array}{|c|} \hline \\ \hline \\ \hline \end{array} \quad
   \begin{array}{|c|} \hline \\ \hline \\ \hline \end{array}
 & 2\\[4mm]
 \multicolumn{2}{|c|}{{\Big\downarrow} \jb} &\quad
 {\Big\downarrow} - &\\[3mm]
   \begin{array}{r|c|l} \cline{2-2} 0& * &0 \\ 
                        \cline{2-2}  & &0 \\ \cline{2-2} \end{array}
 & \begin{array}{r|c|l} \cline{2-2} 0& * &0 \\ \cline{2-2} \end{array}
 & \begin{array}{|c|} \hline \\ \hline \end{array} \quad
   \begin{array}{|c|} \hline \\ \hline \\ \hline \end{array} \quad
   \begin{array}{|c|} \hline \\ \hline \end{array}
 & 3\\[4mm]
 \multicolumn{2}{|c|}{{\Big\downarrow} \jb} &\quad
 {\Big\downarrow} - &\\[3mm]
   \begin{array}{r|c|l} \cline{2-2} 0& &0 \\ \cline{2-2} \end{array}
 & \qquad\emptyset
 & \begin{array}{|c|} \hline \\ \hline \end{array} \quad
   \begin{array}{|c|} \hline \\ \hline \\ \hline \end{array}
 & 1\\[4mm]
 \multicolumn{2}{|c|}{{\Big\downarrow} \jb} &\quad
 {\Big\downarrow} - &\\[3mm]
   \begin{array}{r|c|l} \cline{2-2} 0& * &0 \\ \cline{2-2} \end{array}
 & \qquad\emptyset
 & \begin{array}{|c|} \hline \\ \hline \end{array} \quad
   \begin{array}{|c|} \hline \\ \hline \end{array}
 & 2\\[4mm]
 \multicolumn{2}{|c|}{{\Big\downarrow} \jb} &\quad
 {\Big\downarrow} - &\\[3mm]
   \qquad\emptyset
 & \qquad\emptyset
 & \begin{array}{|c|} \hline \\ \hline \end{array}
 & 1\\[4mm]
 \multicolumn{2}{|c|}{{\Big\downarrow} \jb} &\quad
 {\Big\downarrow} - &\\[3mm]
   \qquad\emptyset
 & \qquad\emptyset
 & \emptyset
 & \\[4mm]
\hline
\end{array}
\end{equation*}
\end{center}
\caption{Example of the bijection\label{tab}}
\end{table}

Here we give an example illustrating the bijection of
Definition-Proposition~\ref{def bij}. Set $\la=(3,2)$ and
$R=((1),(2,2))$. Then $(\nu,J)$ with $\nu=((1),(1))$ and
$J=((0),(0))$ is a $(\la^t;R^t)$-configuration.
In Table~\ref{tab} the sequence of steps under $\jht$ and $\jb$
is given. The vacancy numbers are indicated to the left of the
corresponding part of the partition and the riggings are given on
the right. The selected singular strings in each step are indicated
by $*$. The LR tableau which corresponds to $(\nu,J)$
under the bijection $\phib_R$ can be
obtained as follows. First insert $|\la|=5$ into column $\rkb(\nu,J)=2$
of the shape $\la$. This yields
\begin{equation*}
\begin{array}{|c|c|c|} \hline \;&\;&\; \\ \hline
 \; & 5 & \multicolumn{1}{l}{}\\
\cline{1-2} \end{array}.
\end{equation*}
Next insert 4,3,2,1 into columns 3,1,2,1, respectively, where the second
set of numbers are those of the last column in Table~\ref{tab}.
This yields the LR tableau
\begin{equation*}
\begin{array}{|c|c|c|} \hline 1&2&4 \\ \hline
 3& 5 & \multicolumn{1}{l}{}\\
\cline{1-2} \end{array}.
\end{equation*}

\newpage

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\end{document}

