%%% A LaTeX2e file for a 7 page document %%%%%%%%%%% \documentclass[12pt]{article} \usepackage{amssymb} \usepackage{latexsym} \usepackage{amsmath} \pagestyle{myheadings} \markright{\sc the electronic journal of combinatorics 6 (1999), \#R39\hfill} \thispagestyle{empty} \textwidth=158mm \textheight=217mm \oddsidemargin=0in \evensidemargin=0in \topmargin-.5cm \baselineskip1truecm %\parskip.2truecm %\renewcommand{\baselinestretch}{1.0} \catcode`\@=11 \font\twelvemsb=msbm10 scaled 1200 \font\tenmsb=msbm10 \font\ninemsb=msbm9 \newfam\msbfam \textfont\msbfam=\twelvemsb \scriptfont\msbfam=\tenmsb \scriptscriptfont\msbfam=\ninemsb \def\msb@{\hexnumber@\msbfam} \def\Bbb{\relax\ifmmode\let\next\Bbb@\else \def\next{\errmessage{Use \string\Bbb\space only in math mode}}\fi\next} \def\Bbb@#1{{\Bbb@@{#1}}} \def\Bbb@@#1{\fam\msbfam#1} \catcode`\@=12 \begin{document} %%%% Macros. %%%%%% \def\cara#1{\chi^{#1}} \def\caraper#1{\phi^{#1}} \def\gruposim#1{{\sf S}(#1)} \def\coefi{c(\lambda,\mu,\nu)} \def\produ{\cara \lambda \otimes \cara \mu} \def\particion#1#2{{#1}({#2})} \def\coeficiente#1{c\left( \lambda(#1), \mu(#1), \nu(#1)\right)} \def\lrtablas#1{lr(\lambda,\mu;#1)} \def\liri{{\rm Littlewood-Richardson}\ } \def\domina{\unrhd} \def\internu#1{I_{\nu(#1)}} \def\multitabla#1{{\sf LR}(\lambda(#1),\mu(#1);\nu(#1))} \def\coef#1{c(\lambda,\mu,{#1})} \def\coefvec{c(\lambda,\mu)} \def\kostka#1#2{K_{{#1}{#2}}} \def\kostkafijo{K_{\lambda\mu}} \def\kostkainv#1#2{K^{(-1)}_{{#1}\,{#2}}} \def\kostkainvfijo{K^{(-1)}_{\lambda\mu}} \def\indu#1#2{{\rm Ind}_{#1}^{#2}} \def\ro1t{(\rho(1), \dots ,\rho(t))} \def\ropi{\rho(1)\vdash\pi_1,\dots, \rho(t)\vdash\pi_t} \def\coefgen#1#2{c_{#2}^{#1}} \def\coefgenfijo{\coefgen {\lambda}{\ro1t}} \def\coefgenfijom{\coefgen {\mu}{\ro1t}} \def\partipi{\pi=(\pi_1,\dots, \pi_t)} \def\lintm{\lambda\cap\mu} \def\contri#1#2{{\rm c}_{#1}(\lambda,\mu,{#2})} \def\contrid#1#2{{\rm d}_{#1}(\lambda,\mu,{#2})} \def\vector#1#2{({#1}_2,\dots,{#1}_{#2})} \def\flecha{\longrightarrow} \begin{centering} {\Large\bf Stability of Kronecker products of irreducible characters of the symmetric group}\\[.5cm] {\sf Ernesto Vallejo\footnotemark[1]} \\ Instituto de Matem\'aticas\\ Universidad Nacional Aut\'onoma de M\'exico\\ Area de la Inv. Cient. 04510 M\'exico, D.F.\\ {\small\texttt {evallejo@matem.unam.mx}}\\[.1in] Submitted: October 30, 1998; Accepted: September 6, 1999\\[.1in] Primary classification 05E10, secondary classification 20C30\\ \end{centering} \footnotetext[1]{Supported by DGAPA, UNAM IN103397} \vspace{7mm} \centerline{\bf Abstract} {\small F. Murnaghan observed a long time ago that the computation of the decompositon of the Kronecker product $\cara {(n-a, \lambda_2, \dots )}\otimes \cara {(n-b, \mu_2, \dots)}$ of two irreducible characters of the symmetric group into irreducibles depends only on $\overline\lambda=(\lambda_2,\dots )$ and $\overline\mu =(\mu_2,\dots )$, {\em but not} on $n$. In this note we prove a similar result: given three partitions $\lambda$, $\mu$, $\nu$ of $n$ we obtain a lower bound on $n$, depending on $\overline\lambda$, $\overline\mu$, $\overline\nu$, for the stability of the multiplicity $\coefi$ of $\cara\nu$ in $\produ$. Our proof is purely combinatorial. It uses a description of the $\coefi$'s in terms of signed special rim hook tabloids and \liri multitableaux.} \vskip 3pc {\large\bf 1 Introduction.} \vskip 1.5pc Let $\cara \lambda$ denote the irreducible complex character of the symmetric group $\gruposim n$ corresponding to the partition $\lambda$. For any three partitions $\lambda$, $\mu$, $\nu$ of $n$ we denote by \begin{equation} \coefi := \langle \produ, \cara\nu \rangle \label{definicion} \end{equation} the multiplicity of $\cara\nu$ in the Kronecker product $\produ$. F. Murnaghan observed in \cite{mur} that the computation of the decompositon of the Kronecker product $\cara {(n-a, \lambda_2, \dots )}\otimes \cara {(n-b, \mu_2, \dots)}$ into irreducibles depends only on $\overline\lambda=(\lambda_2,\dots )$ and $\overline\mu =(\mu_2,\dots )$, {\em but not} on $n$. He gave fifty eight formulas for decompositions of Kronecker products corresponding to the simplest choices of $\overline \lambda$ and $\overline\mu$. In fact, his formulas are {\em valid for arbitrary n} only if one follows some rules to restore and discard disordered partitions appearing in them, see comment on \cite[p.762]{mur}. In this note we prove a similar result: given three partitions $\lambda$, $\mu$, $\nu$ of $n$ we obtain a lower bound on $n$, depending on $\lambda$, $\mu$, $\nu$, for the stability of the coefficients $\coefi$. More precisely. Let $\overline\lambda =\vector \lambda p$, $\overline\mu=\vector \mu q$, $\overline\nu=\vector \nu r$ be partitions of positive integers $a$, $b$, $c$ respectively. For each $n\ge a+\lambda_2$ we consider the partition of $n$, $\particion \lambda n := (n-a, \lambda_2,\dots, \lambda_p)$. Similarly we define $\particion \mu n$, and $\particion\nu n$. Then we have \bigskip {\bf Main Theorem.} {\em If\, $\overline \nu$ has one part and $\overline\lambda =\overline\mu$, let $m=\max\{ \lambda_2+a+c,2c\}$; otherwise let $m=\max\{\lambda_2+a+c-1,\mu_2+b+c-1,2c\}$. Then for all $n\ge m$ \[ \coeficiente n= \coeficiente m. \] } \\[-1mm] \indent We note that $m$ is not symmetric on $\lambda$, $\mu$, $\nu$, but $\coefi$ is. Therefore we may have three different choices for $m$ and we choose the smallest of the three. For example, consider partitions $(1)$, $(2,1)$, $(2,1)$. If we set $\overline\lambda=(1)$, $\overline\mu=(2,1)$, $\overline\nu=(2,1)$, then $a=1$, $b=3$, $c=3$ and $m=\max\{4,7,6\}=7$. However, if we set $\overline\lambda=(2,1)$, $\overline\mu=(2,1)$, $\overline\nu =(1)$, then $a=3$, $b=3$, $c=1$ and $m=\max\{6,2\}=6$, and we get a sharper lower bound. This is the best possible, since $c\left( (3,2,1), (3,2,1), (5,1)\right)=2$ and $c\left( (2,2,1), (2,2,1), (4,1)\right)=1$. We also note that the theorem does not always produce the best lower bound. For the partitions $(3,2)$, $(2,2,1)$, $(2,2)$ the lower bound given by the theorem is 11. However, using SYMMETRICA \cite{keko}, we obtained \begin{align*} c((4,3,2), (4,2,2,1), (5,2,2)) &= 12 \\ c((5,3,2), (5,2,2,1), (6,2,2)) &= 16 \\ c((6,3,2), (6,2,2,1), (7,2,2)) &= 16, \end{align*} which shows that the best lower bound is 10. \medskip The rest of this note is devoted to the proof of the theorem. \vskip 3pc {\large\bf 2 Notation, definitions and known results} \vskip 1.5pc In this section we fix the notation and record some definitions and results that will be used in the proof of the theorem. Let $\lambda$ be a partition of $n$, in symbols $\lambda\vdash n$. We denote by $|\lambda|$ the sum of its parts, and by $\lambda '$ its conjugate. We say that $\mu$ is contained in $\lambda$, in symbols $\mu\subseteq\lambda$, if $\mu_i\le\lambda_i$ for all $i$. We use the notation $\lambda\domina \mu$ to indicate that $\lambda$ is greater or equal than $\mu$ in the dominance order. We denote by ${\cal P}(n)$ the {\em diagram lattice}, that is the set of partitions of $n$ together with the dominance order, see \cite{bry, jake, mac, sag}. Let $H$ be a subgroup of a group $G$. If $\chi$ is a character of $H$ we denote by ${\rm Ind}_H^G(\chi)$ the induction character of $\chi$. For any vector $\pi=(\pi_1,\dots,\pi_t)$ of non-negative integers such that $\pi_1 +\cdots +\pi_t=n$, let $\gruposim \pi$ denote a Young subgroup of $\gruposim n$ corresponding to $\pi$. We denote by $\cara\lambda$ the irreducible character of $\gruposim n$ associated to $\lambda$, and by $\caraper \lambda= \indu{\gruposim \lambda}{\gruposim n}(1_\lambda)$ the permutation character associated to $\lambda$. They are related by the Young's rule \begin{equation} \caraper \mu = \sum_{\lambda \unrhd \mu} \kostka\lambda\mu\,\cara\lambda, \label{young} \end{equation} where $\kostka \lambda\mu$ is a Kostka number, that is, the number of semistandard tableaux of shape $\lambda$ and content $\mu$, see \cite[2.8.5]{jake}, \cite[\S 2.11]{sag}. We will deal with two kinds of products of characters: Let $l$, $m$ be non-negative integers, and let $n=l+m$. Let $\chi_1$ be a character of $\gruposim l$, $\chi_2$ be a character of $\gruposim m$, then (i) $\chi_1\times\chi_2$ denotes the character of $\gruposim l\times \gruposim m$ given by $\chi_1\times\chi_2(\sigma, \tau)= \chi_1(\sigma) \chi_2(\tau)$. \vspace{-.3mm} (ii) $\chi_1\otimes\chi_2$ denotes, if $l=m$, the Kronecker product of $\chi_1$ and $\chi_2$, that is, the character of $\gruposim l$ defined by $\chi_1\otimes\chi_2 (\sigma)= \chi_1(\sigma)\chi_2(\sigma)$. If $T$ is a tableau (a skew diagram filled with positive integers) there is a word $w(T)$ associated to $T$ given by reading the numbers of $T$ from right to left, in succesive rows, starting with the top row. Let $\pi=(\pi_1,\dots, \pi_t)$ be a vector of positive integers such that $\pi_1+ \cdots + \pi_t =n$. Let $\rho(i)\vdash\pi_i$, $1\le i\le t$. A sequence $T=(T_1, \dots , T_t)$ of tableaux is called a {\em \liri multitableau} of {\em shape} $\lambda$, {\em content} $\ro1t$ and {\em type} $\pi$ if (i) There exists a sequence of partitions \[ 0=\lambda(0)\subset\lambda(1)\subset\cdots\subset\lambda(t)=\lambda \] such that $|\lambda(i)/\lambda(i-1)|=\pi_i$ for all $1\le i \le t$, and \vspace{-.3mm} (ii) for all $1\le i\le t$, $T_i$ is a semistandard tableau of shape $\lambda(i)/\lambda(i-1)$ and content $\rho(i)$ such that $w(T_i)$ is a lattice permutation, see \cite[2.8.13]{jake}, \cite[I.9]{mac}, \cite[\S 4.9]{sag}. For each partition $\lambda$ of $n$ let $\coefgenfijo$ denote the number of Littlewood-Richardson multitableaux of shape $\lambda$ and content $\ro1t$. It follows by induction from the \liri rule that \[ {\rm Ind}_{\gruposim \pi}^{\gruposim n} \left( \cara{\rho(1)}\times \cdots \times \cara{\rho(t)} \right)= \sum_{\lambda \vdash n} \coefgenfijo \cara\lambda. \] Let \begin{equation}\label{tablas} \lrtablas \pi := \langle \produ, \caraper \pi \rangle, \end{equation} then it follows from the Frobenius reciprocity theorem that \[ \lrtablas \pi= \sum_{\ropi}\coefgenfijo\,\coefgenfijom . \] That is, $lr(\lambda,\mu;\pi)$ is the number of pairs $(S,T)$ of \liri multitableaux of shape $(\lambda,\mu)$, same content, and type $\pi$. Let $K_n=(\kostkafijo)$ be the Kostka matrix with rows and columns arranged in reverse lexicographical order, and let $K_n^{-1} =( \kostkainvfijo )$ denote its inverse, see \cite[I.6.5]{mac}. Then it follows from the Young rule (\ref{young}) that \begin{equation}\label{younginv} \cara\nu= \sum_{\pi\domina\nu}\, \kostkainv \pi\nu \caraper\pi. \end{equation} Therefore from (\ref{definicion}), (\ref{younginv}) and (\ref{tablas}) we obtain \bigskip {\bf 2.1 Proposition} \[ \coefi= \sum_{\pi\domina\nu} \kostkainv \pi\nu \, \lrtablas \pi. \] \hfill $\Box$ \medskip This formula gives, together with a result of E\~gecio\~glu and Remmel \cite{egre} (see Theorem 3.2), a combinatorial description of the numbers $\coefi$. We will use it to get the stability of $\coefi$ from the stability of $\kostkainv \pi\nu$ and $\lrtablas\pi$. \vskip 3pc {\large\bf 3 Proof of the main theorem} \vskip1.5pc Let ${\cal P\/}(n)$ denote the diagram lattice, that is, the lattice of partitions of $n$ ordered under the dominance order, see \cite{bry, jake, mac, sag}. For each partition $\nu$ of $n$, let $I_\nu$ denote the interval $\{\pi \vdash n \mid \nu \unlhd \pi \unlhd (n) \}$ in ${\cal P}(n)$. \bigskip {\bf 3.1 Lemma.} {\em Let $n\ge 2c$. Then the intervals $\internu n$ and $\internu {2c}$ are isomorphic as posets.} {\bf Proof.} For $\pi=(\pi_1,\dots,\pi_t) \in \internu n$ we define $\widetilde\pi := (\pi_1-(n-2c), \pi_2,\dots, \pi_t)$. It follows from the inequality $\pi\unrhd\nu(n)$ that $\widetilde\pi$ is in $\internu {2c}$. One can then easily verify that the map $\pi\mapsto\widetilde \pi$ is a poset isomorphism from $\internu n$ to $\internu {2c}$. \hfill $\Box$ \medskip In fact $2c$ is the best lower bound: Choose $\overline\nu$ be any partition of $c$ with more than one part. Then $\nu(2c)$ and $\nu(2c-1)$ are well defined partitions, but $\internu{2c}\not\cong\internu{2c-1}$, because the partition $(c,c)\in\internu{2c}$ has no corresponding partition in $\internu{2c-1}$. \medskip Next we prove a stability property for the numbers $\kostkainv \lambda\mu$. For this we use a combinatorial interpretation of these numbers due to E\~gecio\~glu and Remmel \cite{egre}. Recall that a {\em special rim hook tabloid} $T$ of shape $\mu$ and type $\lambda$ is a filling of the Ferrers' diagram of $\mu$ with rim hooks of sizes $\{\lambda_1,\dots,\lambda_p\}$ such that each rim hook is {\em special}, that means, each rim hook has at least one box in the first column. The {\em sign} of a rim hook $H$ is $(-1)^{{\rm ht}(H)-1}$, where ${\rm ht(H)}$ denotes, as usual, the height of the rim hook. And the sign of $T$ is defined as the product of the signs of the rim hooks of $T$, see \cite[Section 2]{egre}, \cite[Ex. I.6.4]{mac} for details. Then \bigskip {\bf 3.2 Theorem.} {\em (E\~gecio\~glu, Remmel \cite{egre}) \[ \kostkainv \lambda\mu = \sum_T {\rm sign}(T) , \] where the sum is over all special rim hook tabloids of type $\lambda$ and shape $\mu$.} \bigskip {}From this we get the following two corollaries \medskip {\bf 3.3 Corollary.} {\em Let $\overline\nu=\vector \nu r \vdash c$, and $n\ge 2c$. Then for all $\alpha(n)$, $\beta(n)$ in $\internu n$ one has \[ \kostkainv {\alpha(n)}{\beta(n)}=\kostkainv {\alpha(2c)}{\beta(2c)}. \] } \\[-2.0mm] {\bf Proof.} A sign preserving bijection between the set of special rim hook tabloids $T$ of type $\alpha(2c)$ and shape $\beta(2c)$ and the set of special rim hook tabloids $\widehat T$ of type $\alpha(n)$ and shape $\beta(n)$ is established in the following way: Let $H$ be the rim hook in $T$ which contains the last box from the first row. Then $H$ is of maximal length among the rim hooks in $T$. Let $\widehat H$ be the rim hook obtained from $H$ by adding $n-2c$ boxes at the end of the first row, and let $\widehat T$ be obtained from $T$ by substituting $H$ by $\widehat H$. Since $H$ is of maximal length, then $\widehat T$ is a rim hook tabloid of type $\alpha(n)$. Clearly it has shape $\beta(n)$ and ${\rm sign}(T) ={\rm sign}(\widehat T)$. \hfill $\Box$ \medskip Another proof follows from \cite[Ex. I.6.3]{mac}. \bigskip {\bf 3.4 Corollary.} {\em Let $\overline\nu =(\nu_1,\dots,\nu_r)$, $\overline\pi = (\pi_2,\dots,\pi_t)\vdash c$, and suppose $r>2$. Then \[ \kostkainv {\pi(2c)}{\nu(2c)}=\kostkainv {\overline\pi}{\overline\nu}. \] \hfill$\Box$} \smallskip Since the sum of the entries of any column of the inverse Kostka matrix (with the obvious exception of the first one) is zero, then it follows \bigskip {\bf 3.5 Corollary.} {\em Let $m=2c$, and suppose $r>2$. Then \[ \sum_{\pi(m)\domina\nu(m),\ \pi(m)_1=c} \kostkainv {\pi(m)}{\nu(m)} =0. \] \hfill $\Box$} \bigskip Let denote $\multitabla n$ the set of pairs $(S,T)$ of \liri multitableaux of shape $(\lambda(n),\mu(n))$, same content and type $\nu(n)$. \bigskip {\bf 3.6 Lemma.} {\em Let $m=\max\{\lambda_2+a,\mu_2+b, \nu_2 +c\}$. Then for all $n\ge m$ there is an injective map \[ \Phi :\multitabla m \flecha \multitabla n. \] } \vspace{-3mm} {\bf Proof.} Let $(S,T)\in \multitabla m$. Let $\widehat S$ be obtained from $S$ by adding $n-m$ 1's at the end of the first row of $S_1$, and shifting $n-m$ places to the right the remaining 1's belonging to the tableaux $S_2,\dots S_r$. Let $\widehat T$ be defined in a similar way. Then $(\widehat S,\widehat T)$ belongs to $\multitabla n$, and the map $\Phi(S,T):=(\widehat S,\widehat T)$ is injective.\\[0mm] \mbox{}\hfill$\Box$ \bigskip {\bf 3.7 Proposition.} {\em Let $m=\max\{\lambda_2+a+c-1, \mu_2+b+c-1, \nu_2+c\}$ if $\overline\lambda\neq \overline\mu$, and $m=\max\{\lambda_2+a+c, \nu_2+c\}$ if $\overline\lambda=\overline\mu$. Then for all $n\ge m$ \[ lr(\lambda(n),\mu(n);\nu(n))=lr(\lambda(m),\mu(m);\nu(m)). \] } \\[-3mm] \indent{\bf Proof.} We show that under our hypothesis, we can define a map \[ \Psi:\multitabla n\flecha\multitabla m \] inverse to $\Phi$. Let $(S,T)$ be in $\multitabla n$, and let $(\rho(1), \dots, \rho(r) )$ be the common content of $S$ and $T$. We define $\widetilde{\rho(1)}:= (\rho(1)_1-(n-m), \rho(1)_2, \dots, \rho(1)_u)$, where $u$ is the length of $\rho(1)$. Note that $\rho(1)\subseteq \lambda(n)$ and that $|\lambda(n)/\rho(1)|=c$. Then, if $\overline\lambda=\overline\mu$, we have that $\rho(1)_1\ge \lambda(n)_1 -c=n-a-c \ge n-m+\lambda_2$. And, if $\overline\lambda\neq\overline\mu$, we have that $\rho(1)_1\ge \lambda(n)_1-(c-1)=n-a-(c-1)\ge n-m+\lambda_2$. Therefore, in both cases, $\rho(1)_1-(n-m)\ge \lambda_2\ge\rho(1)_2$, and $\widetilde{\rho(1)}$ is a partition of $m-c=\nu(m)_1$. Let $\widetilde S$ be obtained from $S$ by deleting the first $(n-m)$ 1's in the first row and shifting to the left the remaining numbers $n-m$ places. In this way $\widetilde S$ is a multitableau of shape $\lambda(m)$, content $(\widetilde{\rho(1)}, \rho(2), \dots, \rho(r) )$ and type $\nu(m)$. Moreover, since $\rho(1)_1 - (n-m)\ge \lambda_2$, $\widetilde S$ is a \liri multitableau. We define in a similar way a \liri multitableau $\widetilde T$ of shape $\mu(m)$, same content as $\widetilde S$ and type $\nu(m)$. It is straightforward to check that the map $(S,T)\mapsto (\widetilde{S},\widetilde{T})$ yields the inverse of $\Phi$. \hfill $\Box$ \bigskip {\bf 3.8 Corollary.} {\em Let $m$ be defined as in Proposition 3.7. Let $\pi(m)= (m-e,\pi_2,\dots,\pi_t)$ be in $\internu m$. Then for all $n\ge m$ \[ lr \left(\lambda(n),\mu(n);\pi(n)\right)= lr\left(\lambda(m),\mu(m);\pi(m)\right). \] } \hfill $\Box$ \bigskip The main theorem now follows from Proposition 2.1, Corollaries 3.3 and 3.8, either if $\overline\lambda\neq\overline\mu$, or if $\overline\lambda=\overline\mu$ and $r=2$. It remains to prove it in the case $\overline\lambda=\overline\mu$ and $r >2$. \bigskip {\bf 3.9 Lemma.} {\em Let $m=\max\{\lambda_2+a+c-1, 2c\}$, $\pi(m)=(m-c,\pi_2,\dots,\pi_t)$ be in $\internu m$, and suppose $r>2$. Then for all $n> m$ \[ lr(\lambda(n),\lambda(n);\pi(n)) = lr(\lambda(m),\lambda(m);\pi(m)) +1. \] } \\[-2mm] {\bf Proof.} Let $(S,T)$ be in ${\sf LR}(\lambda(n),\lambda(n);\pi(n))$, and let $\ro1t$ be the common content of $S$ and $T$. Then, as in the proof of Proposition 3.7, we have that $\rho(1)_1\ge n-a-c$. If $\rho(1)_1 > n-a-c$, then $\rho(1)_1\ge n-a-(c-1)\ge n-m+\lambda_2$. Again, as in the proof of Proposition 3.7, there exists $(\widetilde S, \widetilde T)\in {\sf LR}(\lambda(m),\lambda(m);\pi(m))$, such that $\Phi(\widetilde S, \widetilde T)=(S,T)$. If $\rho(1)_1=n-a-c$, then $\lambda(n)/\rho(1)=(c)$. In this situation, there is exactly one \liri multitableau $R$ of shape $\lambda(n)$ and type $\pi(n)$. It has content $\left( \lambda(n)/(c), (\pi_2), \dots, (\pi_t) \right)$. Therefore the pair $(R,R)$ is the only one in ${\sf LR}(\lambda(n),\lambda(n);\pi(n) )$ which is not in the image of $\Phi$. The claim follows. \hfill $\Box$ \bigskip {\bf 3.10 Corollary.} {\em Let $m=\max\{\lambda_2+a+c-1, 2c\}$, and suppose $r>2$. Then for all $n > m$} \[ c(\lambda(n),\lambda(n),\nu(n)) = c(\lambda(m),\lambda(m),\nu(m)). \] {\bf Proof.} By Proposition 2.1 it is enough to prove \[ \sum_{\pi(n)\domina\nu(n)} \kostkainv{\pi(n)}{\nu(n)}\, lr(\lambda(n),\lambda(n);\pi(n))= \sum_{\pi(m)\domina\nu(m)} \kostkainv{\pi(m)}{\nu(m)}\, lr(\lambda(m),\lambda(m);\pi(m)). \] Note that if $\pi(n)=(n-e,\pi_2,\dots,\pi_t)$ and $e