\documentclass[finalversion]{FPSAC2023}
\articlenumber{59}
\usepackage[utf8]{inputenc}
\usepackage{amsfonts}
\usepackage{genyoungtabtikz}
\usepackage{ytableau}
%\usepackage{thmtools}

\bibliography{IHbib}

%\usepackage{subcaption}
%\usepackage{caption}
\allowdisplaybreaks

\def\ncfigskip{\hskip 1.08truein}
\def\sa{\vskip .125truein}
\def\ses{\ = \ }
\def\ds{\displaystyle}
\def\repatop#1#2{{\genfrac{}{}{0pt}{}{#1}{#2}}}
\def\allowhooks{tunnel hooks}
\def\allowhook{tunnel hook}
\def\allowhookfilling{tunnel hook covering}
\def\allowhookfillings{tunnel hook coverings}
\def\allowcell{tunnel cell}
\def\allowcells{tunnel cells}
\def\Allowhooks{Tunnel hooks}
\def\Allowcells{Tunnel cells}
\def\borderhooks{boundary hooks}
\def\bordercells{boundary cells}
\def\borderhook{boundary hook}
\def\bordercell{boundary cell}
\def\border{boundary}
\def\TunnelCells{Tunnel Cells}
\def\THF{THC}%It would be really easy to make this THC!  
\def\GBPR{GBPR}
\def\BGPR{\GBPR}
\def\tunnelhooks{\allowhooks}

%Mu's for all occasions--since we don't want the exponents any more I got rid of them in
%a selective manner.
%I replaced them by what they were originally--so if we need to go back, it is very easy to do-Ed
\def\mur{\mu}%originally \mu^{(r)}
\def\murmin{\mu}%originally \mu^{(r-1)}
\def\mufive{\mu}%originally \mu^{(5)}
\def\muiplus{\mu}%originally \mu^{(i+1)}
\def\mui{\mu}%originally \mu^{(i)}

\def\trian{\diamond}

\def\h{\mathfrak{h}}
\def\B{\mathfrak{b}}
\def\r{\mathfrak{r}}
\def\f{\mathfrak{f}}
\def\H{\mathfrak{H}}
\def\c{\tau}
\def\n{\xi}
%\def\KM{\mathfrak{K}}
\def\K{\mathbf{K}}
\def\HS{H}
\def\tc{\c_r}
\def\td{t}



%\newtheorem{Theorem}[thm]{Theorem}
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\newtheorem{theorem}[thm]{Theorem}
\newtheorem{ill}[thm]{Illustration}
%\newtheorem{ex}{Example}[Theorem]
%\usepackage{xcolor}
\numberwithin{thm}{section}

%\definecolor{lightblue}{rgb}{.678, .847,.902}
\definecolor{lightblue}{rgb}{.85,.9,.96}
\definecolor{lightpurple}{rgb}{.9,.5,.9}
\definecolor{lightred}{rgb}{.9,.7,.7}
\definecolor{grey}{rgb}{.85,.85,.85}

\def\sign{\epsilon}
\def\calN{\mathcal{N}}
\def\calT{\mathcal{T}}
\def\calB{\mathcal{B}}
\def\Alpha{\mathcal{A}}

\Yboxdimx{30pt}
\Ylinecolour{black}
\Ynodecolour{black!50!black}

\usepackage[colorinlistoftodos]{todonotes}
\newcommand{\sarah}[1]{\todo[size=\tiny,color=green!30]{#1 \\ \hfill --- Sarah}}
\newcommand{\Sarah}[1]{\todo[size=\tiny,inline,color=green!30]{#1
      \\ \hfill --- Sarah}}
%\newcommand{\dominic}[1]{\todo[size=\tiny,color=blue!30]{#1 \\ \hfill --- Ed}}
\newcommand{\Ed}[1]{\todo[size=\tiny,inline,color=blue!30]{#1
      \\ \hfill --- Ed}}
\newcommand{\REF}[1]{\todo[size=\tiny,inline,color=red!30]{#1
      \\ \hfill -Add Ref}}

\DeclareMathOperator{\I}{\mathfrak{S}}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\SSYT}{SSYT}
\def\SYM{\Sym}
\DeclareMathOperator{\NSym}{NSym}
\def\NSYM{\NSym}
\DeclareMathOperator{\QSym}{QSym}
\def\QSYM{\QSym}
\DeclareMathOperator{\spin}{bank}
\def\munu{\mu/\nu}
\DeclareMathOperator{\munuhat}{\mu/\nu^{(r-1)}}
\DeclareMathOperator{\munurplus}{\mu/\nu^{(r)}}
\DeclareMathOperator{\SHF}{THC}
\DeclareMathOperator{\taxi}{taxi}
\DeclareMathOperator{\aug}{aug}

\title{A combinatorial interpretation of the noncommutative inverse Kostka matrix}
\author{Edward E Allen\addressmark{1} \and Sarah Mason\addressmark{1}}
\address{\addressmark{1}Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109}
\keywords{immaculate functions, inverse Kostka, NSym}

\received{\today}
\begin{document}

\abstract{We provide a combinatorial formula for the expansion of immaculate noncommutative symmetric functions into complete homogeneous noncommutative symmetric functions.  To do this, we introduce generalizations of Ferrers diagrams which we call GBPR diagrams.  We define tunnel hooks, which play a role similar to that of the special rim hooks appearing in the E\u{g}ecio\u{g}lu-Remmel formula for the symmetric inverse Kostka matrix.  We extend this interpretation to skew shapes and fully generalize to define immaculate functions indexed by integer sequences skewed by integer sequences.  Finally, as an application of our combinatorial formula, we extend Campbell's results on ribbon decompositions of immaculate functions to a larger class of shapes.}

\maketitle



\section{Background and Introduction}

The ring $\Sym$ of symmetric functions on a set of commuting variables consists of all polynomials invariant under the action of the symmetric group.  Symmetric functions play an important role in representation theory, combinatorics, and other areas of mathematics and the physical and natural sciences.  The Schur function basis for $\Sym$, which can be defined combinatorially as the generating function for semi-standard Young tableaux or algebraically through \emph{Bernstein creation operators}, corresponds to irreducible representations of the symmetric group.  Schur function multiplication corresponds to the cohomology of the Grassmannian~\cite{Ful97,Mac95,Sag01}.


The inverse Kostka matrix is the transition matrix from the Schur basis $\{ s_{\lambda} \}_{\lambda}$ of $\Sym$ to the \emph{complete
homogeneous basis} $\{ h_{\lambda} \}_{\lambda}$.  The \emph{Jacobi-Trudi Formula}~\cite{Mac95} is a determinantal formula $s_{\lambda/\nu}=\det(h_{\lambda_i-i-(\nu_j-j)})$ for decomposing Schur 
functions $s_{\lambda/\nu}$ into complete homogeneous symmetric functions $h_{\lambda}$.  E\u{g}ecio\u{g}lu and Remmel's combinatorical interpretation of the inverse Kostka matrix~\cite{EgeRem90} provides a method for writing Schur functions in terms of complete homogeneous symmetric functions using decompositions of the indexing shapes into \emph{special rim hooks}.    

The Hopf algebra $\NSym$ of noncommutative symmetric functions can be thought of as the free associative algebra $\K \langle H_1, H_2 , \hdots \rangle$ generated by algebraically independent, noncommuting complete homogeneous symmetric functions $H_i$ over a fixed commutative field $\K$ of characteristic zero.  Set $H_a:=0$ if $a$ is a negative integer and $H_0:=1$. 
 The function $H_{(\alpha_1, \alpha_2, \hdots, \alpha_{\ell})} = H_{\alpha_1}H_{\alpha_2} \cdots H_{\alpha_{\ell}}$ maps to the complete homogeneous symmetric function
$h_\alpha$ under the ``forgetful map'' from $\NSym$ to $\Sym$.  

The creation operator construction of the Schur functions can be extended to $\NSym$ to produce a new basis for $\NSym$ called the \emph{immaculate basis}~\cite{BBSSZ14}, corresponding to indecomposable modules of the $0$-Hecke algebra~\cite{BBSSZ15}.  We use the equivalent Jacobi-Trudi style formula~\cite{BBSSZ14} to define the immaculate basis and we also extend this construction to introduce the \emph{skew immaculate functions}.  


\begin{defn}{\label{Def:Imunu}}
Let $\mu, \nu \in \mathbb{Z}^k$ be sequences of integers.  Define $(M_{\munu})_{i,j}=H_{(\mu_i-i)-(\nu_j-j)}$. Then $\I_{\munu}=\det(M_{\munu}),$
where the noncommutative determinant in $\NSym$ is obtained by performing the Laplace expansion row by row, starting from the top row.
\end{defn}

%\begin{ex}
%With $\mu=(5,3,2)$ and $\nu=(3,2,1)$, then
%$$\begin{aligned}
%    \I_{\munu}=&\det(M_{\munu})=
%    \det\left(\begin{matrix}
%    H_2&H_4&H_6\\
%    H_{-1}&H_1&H_3\\
%    H_{-3}&H_{-1}&H_1
%    \end{matrix}\right)\\
%    =&H_2H_1H_1+H_4H_3H_{-3}+H_6H_{-1}H_{-1}-%H_2H_3H_{-1}-H_4H_{-1}H_1-H_6H_1H_{-3}\\
%    =&H_2H_1H_1
%\end{aligned}$$
%since $H_0=1$ and $H_{a}=0$ for $a<0.$
%\end{ex}

%\Sarah{I think the example above is wrong, and that pair produces too many negatives.  I propose we replace it with:}

\begin{ex}
With $\mu=(4,5,2)$ and $\nu=(3,1,1)$, then
$$\begin{aligned}
    \I_{\munu}=&\det(M_{\munu})=
    \det\left(\begin{matrix}
    H_1&H_4&H_5\\
    H_1&H_4&H_5\\
    H_{-3}&H_0&H_1
    \end{matrix}\right)\\
    =&H_1H_4H_1-H_1H_5H_{0}-H_4H_1H_1+H_4H_5H_{-3}+H_5H_1H_0-H_5H_4H_{-3}\\
    =&H_{1,4,1}-H_{1,5}-H_{4,1,1}+H_{5,1}
\end{aligned}$$
since $H_0=1$ and $H_{-1}=0.$
\end{ex}

In this extended abstract, we provide a combinatorial formula for the expansion of the immaculate basis into the complete homogeneous basis for $\NSym$ using ribbon-like objects we call \emph{\allowhooks}, generalizing the combinatorial inverse Kostka formula to $\NSym$. 
 A complete discussion of these results including detailed proofs and additional examples can be found in~\cite{AllMas22}.  

\begin{thm}{\label{T:mainskew}\label{T:main}}
The decomposition of the skew immaculate noncommutative symmetric functions $\I_{\munu}$ (with $\mu \in\mathbb{Z}^k$ and $\nu$ a partition with at most $k$ parts, possibly the empty partition) into the
complete homogeneous noncommutative symmetric functions is given by the following formula.
\begin{equation}
\I_{\munu}=\sum_{\gamma\in \SHF_{\munu}} \prod_{r=1
}^k \ \ \sign(\h(r,\c_r))\ H_{\Delta(\h(r,\c_r))},
\label{Eq:TH11}
\end{equation}
where $\SHF_{\munu}$ denotes the \allowhook~coverings of a diagram of shape $\munu,$ and a sign \linebreak $\sign(\h(r,\c_r))$ and integer value $\Delta(\h(r,\c_r))$ are assigned to each \allowhook~$\h(r,\c_r)$ in each $\SHF$ $\gamma$.
\end{thm}

\noindent
Note that the product 
$\prod_{r=1
}^k \ \sign(\h(r,\c_r))\ H_{\Delta(\h(r,\c_r))}$ 
in Eq. \eqref{Eq:TH11} is taken in order from $r=1$ to $k$ since the functions $H_{\Delta(\h(r,\c_r))}$ do not commute.  


Loehr and Niese recently published a combinatorial interpretation of the immaculate inverse Kostka matrix~\cite{LoeNie21}.  Their approach uses transitive tournaments and recursively defined sums, which is quite different from our diagrammatic approach.  Loehr and Niese also provide a diagrammatic method for computing the decomposition of an immaculate function into the complete homogeneous basis when the indexing shape is a partition. Our diagrammatic approach works for all indexing shapes, including all sequences of integers.  Our decomposition can be determined directly by looking at the diagram and recording the values associated to the \allowhooks.


\section{\GBPR~diagrams and \allowcells}{\label{Sec:gbpr}}

Let $\mu=(\mu_1,\ldots,\mu_k)$ be a sequence  and $\nu=(\nu_1,\ldots,\nu_k)$ be a partition.  The \emph{Grey-Blue-Purple-Red (\GBPR) diagram} $D_{\mu/\nu}$ for the skew shape ${\mu/\nu}$ is obtained as follows. 

\begin{enumerate}
\item Place $\nu_i$ grey cells in row $i$ of the diagram (for $1 \le i \le k$), working from bottom to top (to stay consistent with French notation).

\item For each $1 \le i \le k$, divide into cases to place red and blue cells into row $i$ as follows.
\begin{enumerate}
\item If $\mu_i >0$ and $\nu_i \le \mu_i,$ place $\mu_i-\nu_i$ blue cells in row $i$ immediately to the right of the grey cells. 

\item If $\mu_i>0$ and $\mu_i < \nu_i,$ place $\nu_i-\mu_i$ red cells in row $i$ immediately to the right of the grey cells.

\item If $\mu_i \le 0,$ place $|\mu_i|+\nu_i$ red cells  in row $i$ immediately to the right of the grey cells.

%\item If $\mu_i = 0,$ place $\nu_i$ red cells in row $i$ immediately to the right of the grey cells.
\end{enumerate}

\item Any cell in the first quadrant not colored grey, red, or blue is \textit{purple}, but we do not typically color these in illustrations since there are infinitely many purple cells.  
\end{enumerate}

See \cref{ex:table} for an example of a \GBPR~diagram.  Although the \GBPR~diagram does not necessarily contain exactly $\mu_i$ cells in row $i$, the value of $\mu_i$ can be determined from the number of grey, blue, and red cells.  Let $a_i$ be the number of grey cells, $b_i$ be the number of blue cells, and $c_i$ be the number of red cells in row $i$ of a \GBPR~diagram. Then $$a_i+b_i-c_i=\mu_i.$$
 

\begin{defn}{\label{def:connected}}
Cells $(p,q)$ and $(r,s)$ are \emph{adjacent} if and only if $|p-r|+|q-s|=1.$  A collection $C$ of cells is \emph{connected} if for any cells $c, d \in C,$
there is a sequence $c=\c_1, \c_2, \ldots, \c_j=d$ where $\c_i\in C$ and $\c_i$ and $\c_{i+1}$ are adjacent for $1 \le i \le j-1.$
Cells $(p,q)$ and $(r,s)$ are \emph{diagonally adjoining} if both $r=p+1$ and $s=q+1$ or both $p=r+1$ and $q=s+1$.
\end{defn}


At times, it will be convenient to construct diagrams for shapes obtained via prefix removal.  Note that any diagram can be considered as a partial diagram by setting $r=0$.

\begin{defn}
Let $r\in\mathbb{Z}_{\ge 0}, \mu \in \mathbb{Z}^k$, and $\nu \in \mathbb{Z}_{\ge 0}^k$ such that $\nu_{r+1} \ge \nu_{r+2} \ge \hdots \ge \nu_k$.  The \emph{partial diagram} $D^{(r)}_{\mu/\nu}$ is obtained by constructing the \GBPR~diagram $D_{(\mu_{r+1}, \hdots , \mu_k)/(\nu_{r+1},  \hdots,  \nu_k)}$ and shifting the resulting diagram up by $r$ rows, so that the first nonempty row is in row $r+1$.
\end{defn}

This construction allows us to compute the \GBPR~diagram for row $r+1$ through row $k$ even when the first $r$ parts of $\nu$ are not necessarily weakly decreasing.

\begin{ex}
Let $\mu=(-3,1,-1,0,3,-2)$ and $\nu=(2,4,1,0,0)$.  Then $D_{\mu/\nu}^{(2)}$ is given by
\vskip .125truein
\vbox{
\Yfillcolour{lightred}
\young({6,1}{6,2},)
\vskip -4.5pt
\Yfillcolour{lightblue}
\young({5,1}{5,2}{5,3},)
\vskip -4.5pt
\Yfillcolour{lightpurple}
\young({4,1},)
\vskip -4.5pt
\Yfillcolour{grey}
\young({3,1}!<\Yfillcolour{lightred}>{3,2}{3,3},)
}
\end{ex}

We now define \emph{\bordercells}, which intuitively are the cells which lie on the boundary of $\nu$.  Boundary cells will be used to construct hooks that will play a role similar to that of \emph{rim hooks} in the combinatorial interpretation of the symmetric inverse Kostka matrix~\cite{EgeRem90}.

\begin{defn}
\label{Def:Bordcells}
Let $\mu \in \mathbb{Z}^k$ and let $r$ be a positive integer such that $1 \le r \le k$.  Let $\nu \in \mathbb{Z}_{\ge 0}^k$ such that $(\nu_r, \nu_{r+1}, \hdots , \nu_k)$ is a partition and $r \le p \le k$.  A cell in location $(p,q)$ 
is a \emph{\bordercell}~of $D_{\mu/\nu}^{(r-1)}$ iff $$\nu_p+1 \le q\le\nu_{p-1}+1,$$
for $p>r$
and
$$\nu_r+1 \le q\le\max\{\nu_r+1, a_r+b_r+c_r\}$$ 
if $p=r$.  
A \emph{\allowcell} $(p,q)$ of $D^{(r-1)}_{\munu}$
is a \bordercell~such that $$q=\nu_p+1.$$  
\end{defn}

\noindent 
The inequality 
$\nu_r+1\le q\le\max\{\nu_r+1, a_r+b_r+c_r\}$ for row $r$ forces the cell $(r,\nu_r+1)$, as well as all red or blue cells in row $r$, to be boundary cells.  Also note that a cell $(p,q)$ with $p<r$ cannot be a boundary cell of $D^{(r)}_{\mu/\nu}$.  Let $\mathcal{B}_{\mu/\nu}^{(r)}$ 
and $\mathcal{T}_{\mu/\nu}^{(r)}$
respectively denote the collections of \bordercells~and \allowcells~of $D_{\mu/\nu}^{(r)}$.  Set $\mathcal{N}_{\mu/\nu}^{(r)}=\mathcal{B}_{\mu/\nu}^{(r)}-\mathcal{T}_{\mu/\nu}^{(r)}.$

\begin{defn}
\label{D:tunhookdef}
Let $\c_r=(p,q) \in \mathcal{T}_{\munu}^{(r-1)}$ be a \allowcell~in $D_{\munu}^{(r-1)}$.  The \emph{\allowhook}~determined by $\c_r$ is the collection $\h(r,\c_r)$ consisting of all boundary cells in rows $r$ through $p$. 
 The cell $\c_r$ (the farthest northwest cell of $\h(r,\c_r)$) is called the \textit{terminal cell} of $\h(r,\c_r)$.  The \emph{sign} of the \allowhook~$\h(r,\c_r)$, denoted by $\sign(\h(r,\c_r)),$ equals $(-1)^{p-r}$.  If cell
$\hat c\in \h(r,\c)$ then we say that $\h(r,\c)$ \emph{covers}  $\hat c$.
\end{defn}

There are many similar objects in the literature such as skew hooks, ribbons, and border strips~\cite{Mac95}, and the rim hooks~\cite{EgeRem90} appearing in the combinatorial interpretation of the inverse Kostka matrix in $\Sym$.  To generalize these combinatorial objects to the $\NSym$ setting, \allowhooks~need to "tunnel" into the diagram instead of remaining on the rim.  

With $\h(r,\c_r)$ a \allowhook~in the partial diagram $D^{(r)}_{\munu}$ terminating at cell $\c=(p,q)$,
set
\begin{equation}
\Delta(\h(r,\c_r))=b_r-c_r + (\nu_r+1-q)+(p-r).
\label{E:Delta}
\end{equation}
Note that in Equation~\eqref{E:Delta}, since $\nu$ is a partition (whose parts might be $0$),
the cell $(p,q) \in {\calT}^{(r)}_{\munu}$ will be weakly west of 
$(r,\nu_r+1)$.

The proof of the following lemma is immediate since there is only one \allowcell~in each row and moving up a row strictly increases the value of $\Delta(\h(r,\c_r))$.  

\begin{lem}{\label{T:UniqueLengths}}
Given a partial diagram $D^{(r)}_{\mu/\nu}$ with $\mu$ a sequence and $\nu$ a partition, for any fixed $j \in \mathbb{Z}$, there is at most one \allowcell~$\c$ such that $j=\Delta(\h(r,\c_r))$.
\end{lem}

The following iterative procedure provides a method for constructing a {\emph \allowhookfilling~($\SHF$)} of the diagram $D_{\mu/\nu}$.  

\begin{proc}{\label{A:THC}}
Consider $\mu=(\mu_1, \mu_2, \hdots , \mu_k) \in \mathbb{Z}^k$ and a partition $\nu=(\nu_1, \nu_2 , \hdots , \nu_k)$.

\begin{enumerate}[leftmargin=1cm]
\item Construct the partial \GBPR~diagram $D^{(0)}_{\mu/\nu}$ of shape $\munu$.  Set $\nu^{(0)}=\nu$.

\item Repeat the following steps, once for each value of $r$ from $1$ to $k$.

\begin{enumerate}

\item Choose a \allowhook~$\h(r,\c_r)$ in $D^{(r-1)}_{\mu/\nu^{(r-1)}}$ and set $\alpha_r:=\Delta(\h(r,\c_r)).$
\item{\label{St:grey}} For each $1 \le i \le k$, let $\eta_i^{(r)}$ be the number of cells in row $i$ of $\h(r,\c_r)$ and let $\nu^{(r)}$ be the partition defined by $\nu_{i}^{(r)}=\nu_{i}^{(r-1)}+\eta_{i}^{(r)}$ for $1 \le i \le k$.


\item{\label{St:gbpr}} Construct the partial \GBPR~diagram $D^{(r)}_{\mu/\nu^{(r)}}$. 

\end{enumerate}
\item Set $\alpha = (\alpha_1, \hdots , \alpha_k)$.  This will become a subscript appearing in the $H$-expansion of $\I_{\mu}$.
\end{enumerate}
\end{proc}

%Step~\ref{St:grey} appends the cells covered by $\h(r,\c)$ to $\nu^{(r-1)}$.  In the diagram $D^{(r)}_{\mu/\nu^{(r)}}$ constructed in Step~\ref{St:gbpr}, the newly appended cells become grey.  Additional red cells are appended to the remaining rows in Step~\ref{St:gbpr}. One additional red cell is appended to row $i$ for every purple cell covered by $\h(r,\c)$ in row $i$ and two additional red cells are appended to row $i$ for every red cell covered by $\h(r,\c)$ in row $i$.

It is not hard to show that if $\nu^{(r)}$ is the sequence of nonnegative integers produced during Step~\ref{St:grey} of \cref{A:THC}, then $(\nu^{(r)}_{r+1}, \nu^{(r)}_{r+2} , \hdots , \nu^{(r)}_k)$ is a partition.

\begin{ex}{\label{ex:table}}
Letting $\mu/\nu=(-3,5,5,0,5,-2,4,6)/(2,1)$,
we construct a \allowhookfilling~of $D_{\munu}.$ 
%\cref{Fig:mu0nu0start} 
First, we give the \GBPR~diagram of $D^{(0)}_{\munu}$ with cells labelled by (row,column).
%=(-4,-2,0,1, 3,6,7)$ 
%with the corresponding location indicated in each cell, respectively.  
We then use a table to provide details for a particular \allowhookfilling. 
Finally, we illustrate this $\THF$ on the \GBPR~diagram itself.  
\vskip .125truein
%For a detailed description of the process, see Appendix 2.
%The corresponding matrix $M_{\munu}$ is
%\begin{equation}
%   M_{\munu}= M^{(1)}=\begin{pmatrix}
%     H_{-4}&H_{-3}&H_{-2}&H_{-1}&H_0&H_1&H_2\cr
%     H_{-3}&H_{-2}&H_{-1}&H_0&H_1&H_2&H_3\cr
%    H_{-2}&H_{-1}&H_0&H_1&H_2&H_3&H_4\cr
 %   H_{-2}&H_{-1}&H_0&H_1&H_2&H_3&H_4\cr
 %   H_{-1}&H_0&H_1&H_2&H_3&H_4&H_5\cr
 %   H_1&H_2&H_3&H_4&H_5&H_6&H_7\cr
 %   H_1&H_2&H_3&H_4&H_5&H_6&H_7
  %      \end{pmatrix}.
   %     \label{E:BigMat}
%\end{equation}

\hbox{
\vbox{
\Yfillcolour{lightblue}
\young({8,1}{8,2}{8,3}{8,4}{8,5}{8,6},)
%\young(\ \ \ \ \ \,)
\vskip -4.5pt
\Yfillcolour{lightblue}
\young({7,1}{7,2}{7,3}{7,4},)
%\young(\ \ \ \ ,)
\vskip -4.5pt
\Yfillcolour{lightred}
\young({6,1}{6,2},)
%\young(\ \ ,)
\vskip -4.5pt
\Yfillcolour{lightblue}
\young({5,1}{5,2}{5,3}{5,4}{5,5},)
%\young(\ \ \ \ \ ,)
\vskip -4.5pt
\Yfillcolour{lightpurple}
\young({4,1},)
%\young(\ ,)
\vskip -4.5pt
\Yfillcolour{lightblue}
\young({3,1}{3,2}{3,3}{3,4}{3,5},)
%\young(\ \ \ \ \ ,)
\vskip -4.5pt
\Yfillcolour{grey}
\young({2,1}!<\Yfillcolour{lightblue}>{2,2}{2,3}{2,4}{2,5},)
%\young(\ !<\Yfillcolour{lightblue}>\ \ \ \ ,)
\vskip -4.5pt
\Yfillcolour{grey}
\young({1,1}{1,2}!<\Yfillcolour{lightred}>{1,3}{1,4}{1,5}{1,6}{1,7})
%\young(\ \ !<\Yfillcolour{lightred}>\ \ \ \ \ )
}}
%\caption{A \GBPR~diagram of shape $(-3,5,5,0,5,-2,4,6)/(2,1)$}
\label{Fig:mu0nu0start}
%\end{figure}

\vskip .125truein
\noindent
The following table records the process of decomposing the \GBPR~diagram $D_{\munu}$ into \allowhooks.  
Each row $r$ indicates the situation before the $r^{th}$ \allowhook~is placed.  Here $\c_r$ is the \allowcell~at which the \allowhook~beginning in the $r^{th}$ row of the partial diagram terminates.  

\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$r$ & $(\mu_r,\mu_{r+1}, \hdots , \mu_k)$  & $\nu^{(r-1)}$ &   $\c_r$ &  
$\Delta(\h(r, \c_r))$\\
\hline
$1$ & $(-3,5,5,0,5,-2,4,6)$ & $(2,1,0,0,0,0,0,0)$ & $(5,1)$  & $-5+6=1$   \\
\hline
$2$ & $(5,5,0,5,-2,4,6)$ & $(7,3,2,1,1,0,0,0)$ & $(2,4)$   & $2+0=2$  \\
\hline
$3$ & $(5,0,5,-2,4,6)$ & $(7,5,2,1,1,0,0,0)$ & $(4,2)$   & $3+2=5$  \\
\hline
$4$ & $(0,5,-2,4,6)$ & $(7,5,5,3,1,0,0,0)$ & $(5,2)$   & $-3+3=0$ \\
\hline
$5$ & $(5,-2,4,6)$ & $(7,5,5,6,4,0,0,0)$ & $(5,5)$  & $1+0=1$ \\
\hline
$6$ & $(-2,4,6)$ & $(7,5,5,6,5,0,0,0)$  & $(8,1)$  & $-2+2=0$ \\
\hline
$7$ & $(4,6)$ & $(7,5,5,6,5,2,1,1)$  & $(8,2)$   & $3+1=4$ \\
\hline
$8$ & $(6)$ & $(7,5,5,6,5,2,4,2)$  & $(8,3)$   & $4+0=4$\\ 
\hline
%\label{Tb:deltas}
\end{tabular}
\end{center}

%\cref{Fig:mu7nu7short} 
\noindent
The following illustration of the $\THF$ of $D^{(0)}_{\mu/\nu}$ depicts all the \allowhooks~at once and omits the step of converting the colors of the cells to grey as they are covered by \allowhooks.

\vskip .1truein

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%\put(0,0){\line(1,1){10}}
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\Yfillcolour{lightblue}
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\put(40,29.5){\line(1,0){30}}
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\put(240,0){$\Delta=1,\sign=1$}
%Next Tunnel
\put(130,16.5){\line(1,0){30}}
\put(180,13){$\Delta=2,\sign=1$}
%Next Tunnel
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\put(70,43){\line(1,0){30}}
\put(180,25.5){$\Delta=5, \sign = -1$}
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\put(130,43){\line(0,1){13}}
\put(70,56){\line(1,0){60}}
\put(210,39){$\Delta=0,\sign=-1$}
%Next Tunnel
\put(155,56){\line(1,0){10}}
\put(180,53){$\Delta=1,\sign=1$}
%Next Tunnel
\put(40,69){\line(1,0){30}}
\put(40,69){\line(0,1){26}}
\put(90,65){$\Delta=0,\sign=1$}
%Next Tunnel
\put(70,82){\line(1,0){60}}
\put(70,82){\line(0,1){13}}
\put(150,77){$\Delta=4, \sign=-1$}
%Next Tunnel
\put(100,94.5){\line(1,0){90}}
\put(210,92){$\Delta=4, \sign=1$}
\end{picture}
\end{ex}


\section{Proof of \cref{T:main} for non-skewed sequences}

In this section, we give an overview of the proof \cref{T:mainskew} when the indexing shape is not a skew shape.  Recall \cref{Def:Imunu}~\cite{BBSSZ14}
states that 
$$\I_\mu=det(M_\mu)=\sum_{\sigma\in S_k}
\sign(\sigma) (M_\mu)_{i,\sigma_i}.
$$  \cref{T:mainskew} provides a combinatorial interpretation of the decomposition of immaculate functions into complete homogeneous functions using~\allowhookfillings.  The main idea behind our proof
is a bijection between \allowhookfillings~and permutations
(\cref{T:perm}).
Furthermore, each \allowhook~is associated to a number equal to the subscript of the corresponding complete homogeneous function appearing in the matrix $M_{\mu}$ (\cref{T:Mrj}).  
Finally, we show that the product of the signs of the \allowhooks~in a \allowhookfilling~is equal to the sign of the corresponding permutation.

The first lemma needed for the proof of the non-skew case of \cref{T:mainskew} provides foundational insight into how \allowcells~are related to the diagonals parallel to the line $y=x$.  Let
\begin{equation}
\mathcal{L}_{j}:=\{(p,q):p-q+1=j\} = \{(j+m,1+m) | m \in \mathbb{Z}^{\ge 0} \}
\label{E:LA}
\end{equation}
be the collection of cells in the $j^{th}$ diagonal of the first quadrant of the plane, for $1 \le j \le k$.  These diagonals (whose properties are described in the following lemma) will correspond to the entries in the permutations used when computing the determinant of the matrix $M_{\mu}$.

\begin{lem}
\label{T:Losecells}
Let $D^{(r-1)}_{\mu/\nu^{(r-1)}}$ be a partial \GBPR~diagram for $\mu \in \mathbb{Z}^k$ and $\nu^{(r-1)} \in \mathbb{Z}^k_{\ge 0}$ such that $(\nu^{(r-1)}_r, \nu^{(r-1)}_{r+1}, \hdots , \nu^{(r-1)}_k)$ is a partition.  Suppose $\tc,\td \in{\calT}^{(r-1)}_{\mu/\nu^{(r-1)}}$, $\tc \ne \td$, and
$\n\in{\calN}^{(r-1)}_{\mu/\nu^{(r-1)}}.$ Furthermore, suppose $\tc =(p_1,q_1),$
$\td =(p_2,q_2),$ and $\n=(p_3,q_3)$.  Finally,
let $\h(r,\tc)$ be a \allowhook~in 
the diagram $D^{(r-1)}_{\mu/\nu^{(r-1)}}.$
Then
\begin{enumerate}[leftmargin=1cm,label=\Alph*.]
    \item $(p_1+1,q_1+1)\in {\calN}^{(r)}_{\mu/\nu^{(r)}}.$
    
    \item If $\h(r,\tc)$ does not cover $\td$ then
     $\td\in{\calT}^{(r)}_{\mu/\nu^{(r)}}$.
     
     \item If $\h(r,\tc)$ covers $\td$ then $(p_2+1,q_2+1)\in{\calT}^{(r)}_{\mu/\nu^{(r)}}.$
     
     \item
     If $\h(r,\tc)$ does not cover $\n$ then $\n\in{\calN}^{(r)}_{\mu/\nu^{(r)}}.$
    
     \item
     If $\h(r,\tc)$ covers $\n$ then $(p_3+1,q_3+1)\in{\calN}^{(r)}_{\mu/\nu^{(r)}}.$
    
\end{enumerate}
%\label{T:cells}
\end{lem}
\noindent

\cref{T:Losecells} implies an algorithm for identifying the cells in
$\mathcal{T}^{(r-1)}_{\mu/\nu^{(r-1)}}$ available to become terminal cells at each step $r$ in the construction of a \THF.  Along the way, we uncover a permutation associated with each \THF.  This algorithm to identify \allowcells~and produce the associated permutation is described below.

\begin{proc}{\label{Pr:perm}}
The following algorithm constructs a sequence of cells (which we will see are terminal cells for a \allowhookfilling) and also produces a permutation associated to each choice of cells.

\begin{enumerate}[leftmargin=1cm]
\item Let $\mathbb{T}_{(0)} = \{(1,1),(2,1),\ldots, (k,1)\}$.

\item Select a cell $\c_1=(p,1)$ from $\mathbb{T}_{(0)}$ and set $$\mathbb{T}^{\{\c_1\}}_{(1)}=\{(2,2),(3,2),\ldots, (p,2),(p+1,1),\ldots,(k,1)\};$$ $\mathbb{T}^{\{\c_1\}}_{(1)}$ is the set constructed from $\mathbb{T}_{(0)}$ by removing $(p,1)$ and adding $(1,1)$ to each of the cells from rows lower than row $p$.

\item Let $\sigma_1=p-1+1=p$.  Note that $\mathcal{L}_{\sigma_1}$ is the diagonal containing $(p,1)$.

\item Repeat the following steps, once for each value of $r$ from $2$ to $k$.

\begin{enumerate}[noitemsep]
\item Select a cell $\c_r=(p_r,q_r)$ from $\mathbb{T}^{\{\c_1, \hdots , \c_{r-1}\}}_{(r-1)}$.

%be the terminal cell selected during step $r$.  
\item{\label{add11}} Construct $\mathbb{T}^{\{\c_1, \hdots , \c_r\}}_{(r)}$ from $\mathbb{T}^{\{\c_1, \hdots , \c_{r-1}\}}_{(r-1)}$ by removing $\c_r$ and adding $(1,1)$ to each of the cells from rows lower than row $p_r$.

\item Set $\sigma_r=p_r-q_r+1$.  Note that $\mathcal{L}_{\sigma_r}$ is the diagonal containing $(p_r,q_r)$.
\end{enumerate}

\item Set $\sigma=(\sigma_1, \sigma_2, \hdots , \sigma_k)$.
%\item Continue until step $k$ of the \THF~construction is complete; at this point $\mathbb{T}_{\mu/\nu^{(k)}}=\emptyset$.
\end{enumerate}
\end{proc}

The following lemma shows that \cref{Pr:perm} provides a method for selecting the terminal cells of a tunnel hook covering of any non-skew shape $\mu$.   Although the cells chosen do not depend on the diagram $\mu$ (provided $\mu$ is non-skew), the value $\Delta$ of each tunnel hook selected will vary based on the diagram $\mu$.  

\begin{lem}{\label{L:diags}}
Let $\mu$ be an arbitrary integer sequence with $k$ parts and $\nu$ be the empty partition.  Let $\mu/\nu^{(r-1)}$ be the shape obtained after $r-1$ iterations of the second step in \cref{A:THC} with terminal cells $\{ \c_1, \c_2, \hdots \c_{r-1}\}$.  Then the cells contained in $\mathbb{T}^{\{\c_1, \hdots , \c_{r-1}\}}_{(r-1)}$ are precisely the \allowcells~in $\mathcal{T}^{(r-1)}_{\mu/\nu^{(r-1)}}$ for $1 \le r-1 \le k$.  Furthermore, the sequence $\sigma=(\sigma_1, \hdots , \sigma_k)$ produced in step (5) of the procedure is a permutation in $S_k$.
\end{lem}

\cref{L:diags} proves that \cref{Pr:perm} is equivalent to the \THF~construction procedure, since selecting a \allowhook~starting in row $r$ can be done by simply selecting its terminal cell.  Every possible terminal cell for a \allowhook~starting in row $r$ (for the diagram $D^{(r-1)}_{\mu/\nu^{(r-1)}}$) is included in $\mathbb{T}^{\{\c_1, \hdots , \c_{r-1}\}}_{(r-1)}$.  

\begin{ex}
Let $\sigma$ be a permutation of length $k=10$ and consider step $i=6$ in a \allowhookfilling~construction of $\mu$.  Assume the first $5$ \allowhooks~have been constructed with terminal cells $\c_1, \hdots , \c_5$ and $\sigma_i \in \{2,3,6,7,9\}$ for $1 \le i \le 5$.  Then the \allowcells~in $\mathbb{T}^{\{\c_1, \hdots , \c_5\}}_{(5)}$ are in diagonals $\mathcal{L}_1$,
$\mathcal{L}_{4}$, $\mathcal{L}_{5}$, $\mathcal{L}_{8}$, and $\mathcal{L}_{10}$, and, moreover, $$\mathbb{T}^{\{\c_1, \hdots , \c_5\}}_{(5)} = \{(6,6),(7,4),(8,4),(9,2),(10,1)\}.$$  If, for example, the cell $(8,4)$ is selected as the terminal cell $\c_6$ for the next \allowhook, then the new collection of \allowcells~will become 
$$\mathbb{T}^{\{\c_1, \hdots , \c_6\}}_{(6)} = \{(7,7),(8,5),(9,2),(10,1)\},$$ since $(1,1)$ is added to the first two entries of $\mathbb{T}^{\{\c_1, \hdots , \c_5\}}_{(5)}$ and $(8,4)$ is removed.
\end{ex}

The following proposition is a direct consequence of \cref{Pr:perm} and \cref{L:diags}.

\begin{prop}
Let $\mu=(\mu_1, \hdots , \mu_k)$ be a sequence.   
There is a bijection between \allowhookfillings~of the \GBPR~diagram for $\mu$ and  permutations $\sigma\in S_k.$
\label{T:perm}
\end{prop}

Additionally, given a permutation $\sigma=(\sigma_1, \sigma_2, \hdots , \sigma_k)$ in one-line notation, consider the
subscripts of the
entries of the submatrix 
of $(M_\mu)_{i,j}=H_{\mu_i-i+j}$
obtained by removing row $i$ and column $\sigma_i$ for $1 \le i \le r-1$.  These subscripts equal the $\Delta$ values of the tunnel hooks in the partial diagram obtained after $r-1$ tunnel hooks (corresponding to $\sigma_1, \sigma_2, \hdots , \sigma_{r-1}$) have been constructed.

%expanding through the entries $(1,\sigma_1), \ldots, (r-1,\sigma_{r-1}).$  These entries are equal to the lengths of the tunnel hooks in the corresponding skew shape.

\begin{lem} 
Let $\mu$ be a sequence of length $k$ and let $M_{\mu}$ be defined as above.  Assume the first $r-1$ \allowhooks~have been constructed so that their terminal cells do not lie in $\mathcal{L}_j$.  Then
$(M_{\mu})_{r,j}=H_{\Delta(\h(r,\c_r))},$
where $\c_r$ is the unique cell in the diagonal $\mathcal{L}_j$ that is also in $\mathbb{T}^{\{\c_1, \hdots , \c_{r-1}\}}_{(r-1)}$.
\label{T:Mrj}
\end{lem}

We now have all the pieces we need to complete the proof of \cref{T:mainskew}.

\begin{proof}[Sketch of \cref{T:mainskew} proof (non-skew case)]
Recall that
$\displaystyle{\I_\mu=det(M_\mu)=\sum_{\sigma\in S_k}
\sign(\sigma) (M_\mu)_{i,\sigma_i}}.$
 \cref{T:perm} gives a bijection between permutations and tunnel hook coverings.
In \cref{T:Mrj} we show that the subscripts on the entries of the matrix $M_{\mu}$ equal the $\Delta$ values for the corresponding \allowhooks.
All that remains is to show that
$\sign(\sigma)$ equals the product of the signs of the \allowhooks.  This is done using a connection between the Lehmer code~\cite{Leh60}, the sign of a permutation, and the number of rows covered by a~\allowhook.
\end{proof}

The following three examples demonstrate how to construct the \allowhookfillings~for three different indexing shapes.  Notice that the $6$ configurations of terminal cells are the same in all three examples, but the shapes of the actual \allowhooks~(as well as the resulting $\Delta$ values) differ based on the initial diagram $\mu$.

\begin{ex}
$\I_{(3,1,3)}=
H_{(3,1,3)}-H_{(3,2,2)}-H_{(4,0,3)}+H_{(4,2,1)}+H_{(5,0,2)}-H_{(5,1,1)}$

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\label{Ex:CompSHF313}
\end{ex}

\begin{ex}
$\I_{(3,0,3)}=
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\label{Ex:CompSHF303}
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$\I_{(3,-1,3)}=
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\put(-5,29){\line(1,0){35}}
\put(30,16){\line(1,0){35}}
\put(55,29){\line(1,0){10}}

\put(80,-1){\mbox{\footnotesize $H_4,\sign=-1$}}
\put(80,13){\mbox{\footnotesize $H_0,\sign=-1$}}
\put(80,26){\mbox{\footnotesize $H_1,\sign=1$}}

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\put(155,4){\line(0,1){26}}
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\end{picture}

\label{Ex:CompSHF3n13}

\end{ex}
}

We now briefly discuss what happens when an immaculate is indexed by a sequence skewed by partition.  First note that the partial \GBPR~diagrams produced during the construction of a tunnel hook covering are \GBPR~diagrams of skew shapes.  Therefore finding the tunnel hook coverings of a shape skewed by a partition amounts to choosing an appropriate initial sequence and collection of initial tunnel hooks and then completing the remaining tunnel hook coverings.  See~\cite{AllMas22} for details.

\section{Immaculate functions indexed by sequences skewed by non-partition shapes}
\label{S:modifynu}

We now extend \cref{T:main} to find the $H$-decomposition of $\I_{\munu}$ for any pair of integer sequences $\mu,\nu \in \mathbb{Z}^k$.  If all parts of $\nu$ are nonnegative, set $\hat \nu=(\nu_1,\nu_2,\ldots,\nu_{p-1},\nu_{p+1}-1,\nu_{p}+1,\nu_{p+2},\ldots,\nu_k)$
and $(M_{\mu/\hat{\nu}})_{i,j}=H_{{(\mu_i-i)-(\hat\nu_j-j)}},$
so that
$(M_{\mu/\nu})_{i,j}=(M_{\mu/\hat{\nu}})_{i,j}$ for $j \notin \{p, p+1\}.$
Furthermore, since $\hat\nu_{p+1}=\nu_p+1$
and $\hat\nu_p=\nu_{p+1}-1$, we have $\I_{\munu}=-\I_{\mu/\hat \nu},$ since we obtain $M_{\mu/\hat{\nu}}$ by swapping columns $p$ and $p+1$ of $M_{\mu/\nu}$.  If the entries of $\nu$ are not weakly decreasing, apply the straightening operator $\hat{\nu}$ (adjusting the sign each time) until the result is a partition (in which case \cref{T:mainskew} applies) or a sequence containing a one-step increase (in which case the skew immaculate function is zero).  If any terms of $\nu$ are negative, let $\nu_j$ be the smallest part of $\nu$.  Add $-\nu_j$ to every part of $\mu$ and every part of $\nu$; call the resulting sequences $\aug_{\nu_j}(\mu)$ and $\aug_{\nu_j}(\nu)$ respectively.  Then $M_{\aug_{\nu_j}(\mu) / \aug_{\nu_j}(\nu)} = M_{\munu}$, so $\I_{\munu}=\I_{\aug_{\nu_j}(\mu) / \aug_{\nu_j}(\nu)}$
and we can apply the above techniques to $\I_{\aug_{\nu_j}(\mu) / \aug_{\nu_j}(\nu)}$ to find the decomposition of $\I_{\munu}$ into the complete homogeneous noncommutative symmetric functions.

\begin{ex}
Let $\mu=(2,-5,0,1)$ and $\nu=(2,-3,1,6)$.  To find $\I_{\munu}$, first add $3$ to every part of $\nu$ and every part of $\mu$ to get $\aug_3(\mu)/\aug_3(\nu) = (5,-2,3,4)/(5,0,4,9).$  Next, since $\aug_3(\nu)$ is not a partition, apply the straightening operator to get 
{\small\[
\I_{\mu/\nu} = -\I_{(5,-2,3,4)/(5,0,8,5)} \\ = \I_{(5,-2,3,4)/(5,7,1,5)} = -\I_{(5,-2,3,4)/(6,6,1,5)} = \I_{(5,-2,3,4)/(6,6,4,2)}.
\]}
\end{ex}

\begin{cor}
For $1 \le m \le k$,
we have
$$
\I_\mu=
\sum_{\pi
\in A_{k,m}}
\sign(\pi)
H_{\mu_1-1+\pi_1}
\cdots
H_{\mu_{m-1}-(m-1)+\pi_{m-1}}
\I_{\mu/\nu^{(m)}},
$$
where $\nu^{(m)}$ is the partition obtained from the construction of the \allowhooks~corresponding to $\pi$ in the diagram $D_{\mu}$ and $A_{k,m}$ is the set of all ordered $m$-element subsets of a $k$-element set.
\end{cor}

For example, the skew immaculate $\I_{(5,-1,3,4)/(3,1,0,0)}$ can be obtained by selecting the terms appearing in $\I_{(3,3,3,5,-1,3,4)}$ whose first three terms are $H_{7} H_{4} H_{2}$.  Applying the forgetful map to this expansion produces the expansion of a Schur function in terms of skew Schur functions with complete homogeneous symmetric functions as coefficients.

\section{Ribbon decompositions of immaculate functions}{\label{Sec:Ribbons}}

Ribbons are a Schur-like basis for $\NSym$ related to Schur functions by the forgetful map; there are a number of excellent sources for background on ribbons in $\Sym$ and $\NSym$~\cite{GKLLRT95,Mac95}.  The ribbon basis expands positively into the immaculate basis via standard immaculate tableaux~\cite{BBSSZ14}, but the expansion of the immaculate basis into the ribbon basis is only known for certain special cases.  Campbell~\cite{Cam17} provides formulas for the ribbon expansion of immaculate functions indexed by rectangles and immaculate functions indexed by products of two rectangles satisfying certain size conditions as follows.  

\begin{thm}{\cite{Cam17}}
The ribbon expansion of an immaculate function indexed by either a rectangle $\alpha=m^{\ell(\alpha)}$ or the product $\alpha=(a^b, c^d)$ of rectangles satisfying $b \le c$ and $b \le a$ is given by $$\I_{\alpha} = \sum_{\sigma \in S_{\ell(\alpha)}} \sign(\sigma) R_{(\alpha_1-1+\sigma_1, \alpha_2-2+\sigma_2, \hdots , \alpha_{\ell(\alpha)}-\ell(\alpha)+\sigma_{\ell(\alpha)})}.$$
\end{thm}

It is not true in general that the ribbon decomposition has the same indexing set as the homogeneous decomposition.  One open question is to classify the compositions for which the indexing compositions are the same.  We use \allowhooks~to extend Campbell's results to a much larger class of compositions.

\begin{theorem}{\label{T:im2rib}}
Let $\alpha=(\alpha_1, \alpha_2, \hdots , \alpha_k)$ be a composition for which there exists $j$ with $1 \le j \le k$ such that
$\alpha_i \ge i$ for $1 \le i \le j,$ $\alpha_{j+1} \ge j,$
and $\alpha_{\ell}=\alpha_{j+1}$ for $j+2 \le {\ell} \le k.$
Then 
$$\I_{\alpha}=\sum_{\sigma \in S_k} \sign(\sigma) R_{(\alpha_1-1+\sigma_1,\alpha_2-2+\sigma_2, \hdots , \alpha_k-k+\sigma_k)},$$ with the convention that $R_{\alpha}$ vanishes if $\alpha$ contains any nonpositive parts.
\end{theorem}

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