%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[final]{FPSAC2022}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

\newtheorem{thm}{Theorem}
\newtheorem{prop}{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{lemma}{Lemma}
\newtheorem{conjecture}{Conjecture}

\theoremstyle{definition}
\newtheorem{definition}{Definition}

\theoremstyle{remark}
\newtheorem*{remark}{Remark}
\newtheorem{example}{Example}


%my commands
\usepackage{tikz, outlines, bm}

\def\force{\operatorname{force}}
\def\area{\operatorname{area}}
\def\ldinv{\operatorname{ldinv}}
\def\pdinv{\operatorname{pdinv}}

%bibliography
\usepackage[backend=bibtex]{biblatex}
\addbibresource{q_model.bib}


%\usepackage{lipsum}

%% define your title in the usual way
\title{The Elliptic Hall Algebra Element $\mathbf{Q}_{m, n}^{k} (1)$}


%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[A.~Wilson]{Andy Wilson\thanks{\href{mailto:awils342@kennesaw.edu}{awils342@kennesaw.edu}}}

%% then use \addressmark to match authors to institutions here
\address{Kennesaw State University, 1000 Chastain Rd NW, Kennesaw, GA 30144, USA}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{
Suppose $M$ and $N$ are positive integers and let $k = \gcd(M, N)$, $m = M/k$, and $n=N/k$. We define a symmetric function $L_{M,N}$ as a weighted sum over certain tuples of lattice paths and conjecture that this function is equal (up to a constant) to the elliptic Hall algebra operator $\mathbf{Q}_{m,n}$ composed $k$ times and applied to 1. We show that $L_{M,N}$ satisfies a generalization of Mellit and Hogancamp's recursion for the triply-graded Khovanov--Rozansky homology of the $M,N$-torus link. As a corollary, we obtain the triply-graded Khovanov--Rozansky homology of the $M,N$-torus link as a specialization of $L_{M,N}$.
}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{lattice paths, link homology, torus links, elliptic Hall algebra}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{sample.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction}

For coprime positive integers $m$ and $n$, much has been discovered in recent years about the relationship between $m,n$-torus knots, the elliptic Hall algebra, and $m,n$-Dyck paths (lattice paths from $(0,0)$ to $(m,n)$ staying above the line $my = nx$). More precisely, the relevant objects are
\begin{enumerate}[label={(\arabic*)}]
\item \label{kr} the triply-graded Khovanov--Rozansky homology of the $m,n$-torus knot,  
\item \label{sym} a certain elliptic Hall algebra operator $\mathbf{Q}_{m,n}$ (defined in Section \ref{sec:hall}) applied to 1, and
\item  \label{cat} a generating function over $m,n$-Dyck paths, weighted by variables $q$ and $t$ as well as monomials in variables $x_1, x_2, x_3 \ldots$.

\end{enumerate}
Gorsky and Negut conjectured that \ref{sym} and \ref{cat} are equal up to a sign \cite{gorsky-negut}. This conjecture was proved by Mellit \cite{mellit-rational}. An earlier result of Gorsky implies that \ref{kr} appears as a certain specialization of \ref{sym} and \ref{cat} \cite{gorsky}. 
%lots of citations

Somewhat less explored, but still fairly well understood, is the case where $m=n$, \textit{i.e.}\ $m$ and $n$ are ``minimally coprime''. In this case, the objects are
\begin{enumerate}[label={(\Roman*)}]
\item \label{kr_} the triply-graded Khovanov--Rozansky homology of the $n,n$-torus link (no longer a knot),
\item \label{sym_} the symmetric function $\nabla p_{1^n}$, where $\nabla$ is the Macdonald eigenoperator, and %scaling factor? 
\item \label{cat_} a generating function over an arbitrary number of boxes placed into $n$ columns \cite{torus-link-wilson}. 
\end{enumerate}
\ref{sym_} and \ref{cat_} are conjectured to be equal \cite{torus-link-wilson}. Although this conjecture is still open, it is possible that recent developments in the area could make progress on this problem. The same specialization as before allows one to move from \ref{sym_} or \ref{cat_} to \ref{kr_}, making use of a recursion of Elias and Hogancamp \cite{elias-hogancamp}.

The goal of this abstract is to generalize both the $\gcd(m, n) = 1$ and $m=n$ cases to any positive integers $M$ and $N$. We let $k = \gcd(M, N)$, $m = M/k$, and $n=N/k$. The objects are now 
\begin{enumerate}[label={(\Alph*)}]
\item \label{kr__} the triply-graded Khovanov--Rozansky homology of the $M,N$-torus link, 
\item \label{sym__} the elliptic Hall algebra operator $\mathbf{Q}_{m,n}$ applied iteratively $k$ times to 1, and
\item  \label{cat__} a generating function over $k$-tuples of (variations of) $m,n$-Dyck paths.
\end{enumerate}
The correct analog of Elias and Hogancamp's recursion for \ref{kr__} is known due to Hogancamp and Mellit \cite{hogancamp-mellit}. This recursion guided much of our progress. We develop \ref{sym__} and \ref{cat__} in Section \ref{sec:hall} and Section \ref{sec:combinatorics}, respectively. In Conjecture \ref{conj:main}, which appears at the end of Section \ref{sec:combinatorics}, we posit a precise relationship between \ref{sym__} and \ref{cat__}. In Section \ref{sec:connections}, we explain what we can prove about connections between these objects and what is still unknown. Our main result in this section (Theorem \ref{thm:L_recursion}) is a lift of Hogancamp and Mellit's recursion to the level of symmetric functions using the path tuples from \ref{cat__}. We also make explicit connections between our work and past work on torus link homology and the Rational Shuffle Theorem.

\section{The elliptic Hall algebra}
\label{sec:hall}

Suppose throughout that $M$ and $N$ are positive integers and let $k = \gcd(M, N)$, $m = M/k$, and $n=N/k$. We let $\Lambda$ denote the ring of symmetric functions in variables $x_1, x_2, x_3, \ldots$ over the ground field $\mathbb{Q}(q, t, a, z)$. We use \emph{plethystic substitution}, \textit{i.e.}\ for any formal sum of monomials $A = a_1 + a_2 + a_3 + \cdots$ from the $x_i$'s or the ground field, we define
$$p_k[A] = a_1^k + a_2^k + a_3^k + \cdots$$
for any power sum polynomial $p_k$. We extend plethystic substitution to all of $\Lambda$ by viewing  the $p_i$'s as algebraically independent generators for $\Lambda$. We also let $$X = x_1 + x_2 + x_3 + \cdots.$$ Our first definition is an important operator on $\Lambda$ in the study of Macdonald polynomials. 

\begin{definition}
For any nonnegative integer $k$ and any symmetric function $f$, 
$$\mathbf{D}_k f[X] = \left. f[X + (1-q)(1-t)/z] \sum_{i \geq 0} (-z)^i e_i[X] \right|_{z^k}, $$
where $\left. \right|_{z^k}$ extracts the coefficient of $z^k$ from the series to its left.
\end{definition}

Next, we define the fundamental operators for the elliptic Hall algebra. 

\begin{definition}
For nonnegative integers $m$ and $n$ and $f \in \Lambda$, we define $\mathbf{Q}_{m, n} f$ by first setting
\begin{align*}
\mathbf{Q}_{0, n} f &= \frac{qt}{qt-1} h_n[(1-qt)X/(qt)] \cdot f \\
\mathbf{Q}_{1, n} f &= \mathbf{D}_n f .
\end{align*}
Otherwise, $m \geq 2$. We assume $m$ and $n$ are coprime, so there are unique integers $1 \leq a < m$ and $1 \leq b < n$ such that $na - mb = 1$. Then we let
$$\mathbf{Q}_{m, n} f = \frac{1}{(1-q)(1-t)} [ \mathbf{Q}_{m-a, n-b}, \mathbf{Q}_{a, b} ] f.$$
\end{definition}


$\mathbf{Q}_{m,n}$ is also defined when $m$ and $n$ are not coprime, but we will not need this level of generality \cite{rational-shuffle}. $\mathbf{Q}_{m,n}$ applied to 1 is the symmetric function piece of the Rational Shuffle Theorem. Our next goal is to develop the combinatorial objects which we will use to propose a formula for $\mathbf{Q}_{m,n}^{k}(1).$ 



\section{Combinatorics}
\label{sec:combinatorics}


Again, for positive integers $M$ and $N$ we let $k = \gcd(M, N)$, $m=M/k$, and $n=N/k$. We will define a generating function as a sum over certain $k$-tuples of lattice paths. These lattice paths will depend on a pair of sequences $v$ and $w$ of lengths $M$ and $N$, respectively, in the alphabet $\{0, 1, \bullet\}$. Furthermore, we let
$$|v| = \text{the number of 1's in $v$} $$
and insist that $|v| = |w|$. We will denote the resulting generating function by $L(v, w)$. When $v = 0^M$ and $w=0^N$, we conjecture that $L(v,w)$ provides a combinatorial model for $\mathbf{Q}_{m,n}^k (1)$. 

\subsection{Path tuples}
\label{ssec:tuples}

\begin{definition}
An $m,n$-\emph{path}, or just a \emph{path},  is a sequence of $n$ unit-length north and $m$ unit-length east steps from $(a,0)$ to $(a+m, n)$ for some integer $a$ that
\begin{itemize}
\item begins with a north step, and
\item stays weakly above the line $my = nx.$
\end{itemize}
\end{definition}

We can imagine repeating the sequence of steps in a path to get an ``infinite'' path in the plane. Any height-$n$ ``band'' (region between $y = b$ and $y = b + n$ for some integer $b$) of the infinite path determines the original path. We will most often work with the original $b=0$ band. Next, we define a labeling of the unit squares in each of $k$ ``sheets'' of $\mathbb{Z}^2$. 

\begin{definition}
We consider $\left(\mathbb{Z}^2\right)^k$ as $k$ \emph{sheets} of $\mathbb{Z}^2$, where the sheets are indexed from $0$ to $k-1$. 
The \emph{content} of a lattice square (or ``cell'') in the $i^{\text{th}}$ sheet, where $0 \leq i < k$, is
$$i + My - Nx,$$
where $(x, y)$ are the coordinates of the lower right lattice point of the square.
\end{definition}

We depict the contents of some cells in Figure \ref{fig:paths}. Content provides a bijective correspondence between the cells in any band of $\left(\mathbb{Z}^2\right)^k$ and $\mathbb{Z}$. As a a result, given a fixed band, we can refer to a cell by its content. In the sequel, we often use the phrase ``cell $c$'' to refer to the unique cell in the current band with content equal to $c$. 

\begin{definition}
For a fixed $k$-tuple of paths $\bm{P}$ in sheets $0, 1, \ldots, k-1$ of $\mathbb{Z}^2$, fix a cell $c$ and consider the path $P$ in $\bm{P}$ that is in $c$'s sheet. We say that cell $c$ 
\begin{itemize}
\item \emph{has a north step in $\bm{P}$} if there is a north step from $P$ on $c$'s left boundary, 
\item \emph{is left of $\bm{P}$} if the cell in $c$'s row that has a north step in $P$ is weakly to the right of $c$, and
\item \emph{is right of $\bm{P}$} if the cell in $c$'s row that has a north step in $P$ is strictly to the left of $c$.
\end{itemize}
Analogously,  cell $c$
\begin{itemize}
\item \emph{has an east step in $\bm{P}$} if there is an east step from $P$ on $c$'s upper boundary, 
\item \emph{is above $\bm{P}$} if the cell in $c$'s column that has an east step from $P$ is weakly below $c$, and
\item \emph{is below $\bm{P}$} if the cell in $c$'s column that has an east step from $P$ is strictly above $c$.
\end{itemize}
\end{definition}

Now we use sequences $v$ and $w$ in the alphabet $\{0, 1, \bullet\}$ of lengths $M$ and $N$, respectively, such that 
$|v| = |w|$
to restrict the set of paths we consider. These sequences $v$ and $w$ govern the relative location of the paths and the cells $0, 1, \ldots, M-1$  and $0, 1, \ldots, N-1$. 

\begin{figure}
\centering
\begin{tikzpicture}[scale=0.5]
\def\M{6}
\def\N{4}
\def\m{3}
\def\n{2}
\def\i{0}
%\filldraw[yellow!30!white] (0,1) rectangle (1,2);
\draw[gray, <->] (-3,0) -- (3,0);
\draw[gray, <->] (0,-1) -- (0,\n+1);
\draw[gray, dashed] (0,0) -- (\m, \n);
\draw[gray, dashed] (-3,2) -- (3,2);

\foreach \x in {-3,...,3} {
	\foreach \y in {0,1,...,\n} {
		\fill[color=gray] (\x,\y) circle (0.1);
        }
    }
\foreach \x in {-2,...,2} {
	\foreach \y in {1,...,\n} {
		\pgfmathsetmacro\c{int(\M*(\y-1) - \N*(\x)+\i)}
		\node at (\x-0.5, \y-0.5) {\small$\c$};
        }
    }
\draw[very thick, blue] (-2,1) -- (-1,1) -- (-1,2) -- (0,2);
\draw[very thick, blue, dashed] (-2,0) -- (-2,1);
\draw[very thick, blue, dashed] (1,2) -- (0,2);


\end{tikzpicture}
\hspace{5pt}
\begin{tikzpicture}[scale=0.5]
\def\M{6}
\def\N{4}
\def\m{3}
\def\n{2}
\def\i{1}
%\filldraw[yellow!30!white] (-3,0) rectangle (0,2);
%\filldraw[yellow!30!white] (-2,1) rectangle (1,2);
\draw[gray, <->] (-3,0) -- (3,0);
\draw[gray, <->] (0,-1) -- (0,\n+1);
\draw[gray, dashed] (0,0) -- (\m, \n);
\draw[gray, dashed] (-3,2) -- (3,2);

\foreach \x in {-3,...,3} {
	\foreach \y in {0,1,...,\n} {
		\fill[color=gray] (\x,\y) circle (0.1);
        }
    }
\foreach \x in {-2,...,2} {
	\foreach \y in {1,...,\n} {
		\pgfmathsetmacro\c{int(\M*(\y-1) - \N*(\x)+\i)}
		\node at (\x-0.5, \y-0.5) {\small$\c$};
        }
    }
\draw[very thick, blue] (-3,0) -- (-3,1) -- (-1,1);
\draw[very thick, blue, dashed] (-1,1) -- (0,1);
\draw[very thick, blue, dashed] (0,1) -- (0,2);
\end{tikzpicture}
\hspace{60pt}
\begin{tikzpicture}[scale=0.5]
\def\M{6}
\def\N{4}
\def\m{3}
\def\n{2}
\def\i{0}
%\filldraw[yellow!30!white] (-2,1) rectangle (-1,2);
\draw[gray, <->] (-3,0) -- (3,0);
\draw[gray, <->] (0,-1) -- (0,\n+1);
\draw[gray, dashed] (0,0) -- (\m, \n);
\draw[gray, dashed] (-3,2) -- (3,2);

\foreach \x in {-3,...,3} {
	\foreach \y in {0,1,...,\n} {
		\fill[color=gray] (\x,\y) circle (0.1);
        }
    }
\foreach \x in {-2,...,2} {
	\foreach \y in {1,...,\n} {
		\pgfmathsetmacro\c{int(\M*(\y-1) - \N*(\x)+\i)}
		\node at (\x-0.5, \y-0.5) {\small$\c$};
        }
    }
\draw[very thick, red] (-2,1) -- (-2,2) -- (0,2);
\draw[very thick, red, dashed] (-2,0) -- (-2,1);
\draw[very thick, red, dashed] (1,2) -- (0,2);


\end{tikzpicture}
\hspace{5pt}
\begin{tikzpicture}[scale=0.5]
\def\M{6}
\def\N{4}
\def\m{3}
\def\n{2}
\def\i{1}
\draw[gray, <->] (-3,0) -- (3,0);
\draw[gray, <->] (0,-1) -- (0,\n+1);
\draw[gray, dashed] (0,0) -- (\m, \n);
\draw[gray, dashed] (-3,2) -- (3,2);

\foreach \x in {-3,...,3} {
	\foreach \y in {0,1,...,\n} {
		\fill[color=gray] (\x,\y) circle (0.1);
        }
    }
\foreach \x in {-2,...,2} {
	\foreach \y in {1,...,\n} {
		\pgfmathsetmacro\c{int(\M*(\y-1) - \N*(\x)+\i)}
		\node at (\x-0.5, \y-0.5) {\small$\c$};
        }
    }
\draw[very thick, red] (-3,0) -- (-3,1) -- (-1,1);
\draw[very thick, red, dashed] (-1,1) -- (0,1);
\draw[very thick, red, dashed] (0,1) -- (0,2);
\end{tikzpicture}
%\hspace{80pt}
%\begin{tikzpicture}[scale=0.5]
%\draw[gray, very thin] (0,0) grid (4,4);
%
%\node at (0.5,0.5) {\small 2};
%\node at (1.5,1.5) {\small 5};
%\node at (2.5,2.5) {\small 7};
%\node at (3.5,3.5) {\small 8};
%
%\draw[very thick, purple] (0,2) -- (0,3) -- (1,4) -- (4,4);
%\end{tikzpicture}
\caption{We have drawn the only two possible path tuples for  $v = 000110$ and $w=0110$, one in blue and one in red. The areas of the path tuples are 0 and 1, respectively. Both tuples have 4 path diagonal inversions: $(3, 6), (4, 6), (4,9), (6,9)$. 
}
\label{fig:paths}
\end{figure}


\begin{definition}
A $v,w$-\emph{path tuple} (or just a \emph{path tuple}) is a $k$-tuple of $m,n$-paths $\bm{P} = (P^{(0)}, P^{(1)}, \ldots, P^{(k-1)})$, one in each sheet, such that, for $v = v_0 \ldots v_{M-1}$,
\begin{enumerate}[label={(\roman*)}]
\item $v_i = 0$ if and only if cell $i$ is right of $\bm{P}$,
\item \label{basement1} $v_i = 1$ if and only if cell $i$ has a north step in $\bm{P}$, and
\item $v_i = \bullet$ if and only if cell $i$ is left of $\bm{P}$.
\end{enumerate}
and, for $w = w_0 \ldots w_{N-1}$, 
\begin{enumerate}[label={(\roman*')}]
\item $w_i = 0$ if and only if cell $i$ is below $\bm{P}$,
\item \label{basement2} $w_i = 1$ if and only if cell $i$ has an east step in $\bm{P}$, and
\item $w_i = \bullet$ if and only if cell $i$ is above $\bm{P}$.
\end{enumerate}
The north (respectively east) steps insisted upon above are called \emph{basement} steps and denoted with dashed lines in our figures.
\end{definition}

As a sanity check, we note that, for any $M$ and $N$, the restriction that paths are weakly above the line $my = nx$ implies that there is exactly one path tuple if $v = \bullet^M$ and $w = \bullet^N$. On the other hand, there are many pairs of sequences for which there are infinitely many path tuples, \textit{e.g.}\ $v=0^M$ and $w=0^N$. We address this issue in Subsection~\ref{ssec:compressed}. Figure~\ref{fig:paths} contains two examples of path tuples. Next, we define statistics for these objects. 



\begin{definition}
The \emph{area} of a path tuple $\bm{P}$ is the number of cells $c \geq M+N$ that are weakly right of $\bm{P}$, \textit{i.e.}\ they are either right of $\bm{P}$ or they have north steps in $\bm{P}$.
\end{definition}

\noindent
Next, we define a notion of diagonal inversion for a path.

\begin{definition}
A \emph{path diagonal inversion} in a path tuple $\bm{P}$ is a pair of cells $c$ and $d$ with $c \geq 0$ and  $M \leq d < c + M$ such that 
\begin{itemize}
\item $c$ has a north step in $\bm{P}$ and
\item $d$ is weakly to the right of $\bm{P}$.
\end{itemize}
$c$ and $d$ may be in different sheets and the north steps mentioned above can be basement steps. We let $\pdinv(\bm{P})$ denote the number of path diagonal inversions in a path tuple $\bm{P}$. 
\end{definition}

%\begin{figure}
%\begin{center}
%\begin{tikzpicture}[scale=0.5]
%\def\M{6}
%\def\N{4}
%\def\m{3}
%\def\n{2}
%\def\i{0}
%\filldraw[yellow!30!white] (0,1) rectangle (1,2);
%\draw[gray, <->] (-1,0) -- (\m+1,0);
%\draw[gray, <->] (0,-1) -- (0,\n+1);
%\foreach \x in {0,1,...,\m} {
%	\foreach \y in {0,1,...,\n} {
%		\fill[color=gray] (\x,\y) circle (0.1);
%        }
%    }
%\foreach \x in {1,2,...,\m} {
%	\foreach \y in {1,2,...,\n} {
%		\pgfmathsetmacro\c{int(\M*(\y) - \N*(\x-1)+\i)}
%		\node at (\x-0.5, \y-0.5) {\small$\c$};
%        }
%    }
%\draw[very thick, blue] (0,0) -- (0,2) -- (3,2);
%\end{tikzpicture}
%\hspace{5pt}
%\begin{tikzpicture}[scale=0.5]
%\def\M{6}
%\def\N{4}
%\def\m{3}
%\def\n{2}
%\def\i{1}
%\filldraw[yellow!30!white] (-3,0) rectangle (0,2);
%\filldraw[yellow!30!white] (-2,1) rectangle (1,2);
%\draw[gray, <->] (-4,0) -- (2,0);
%\draw[gray, <->] (0,-1) -- (0,\n+1);
%\foreach \x in {-3,...,1} {
%	\foreach \y in {0,1,...,\n} {
%		\fill[color=gray] (\x,\y) circle (0.1);
%        }
%    }
%\foreach \x in {-2,...,2} {
%	\foreach \y in {1,2,...,\n} {
%		\pgfmathsetmacro\c{int(\M*(\y) - \N*(\x-1)+\i)}
%		\node at (\x-0.5, \y-0.5) {\small$\c$};
%        }
%    }
%\draw[very thick, blue] (-3,0) -- (-3,1) -- (-2,1) -- (-2,2) -- (0,2);
%
%
%\end{tikzpicture}
%\hspace{50pt}
%\begin{tikzpicture}[scale=0.5]
%\def\M{6}
%\def\N{4}
%\def\m{3}
%\def\n{2}
%\def\i{0}
%\draw[gray, <->] (-1,0) -- (\m+1,0);
%\draw[gray, <->] (0,-1) -- (0,\n+1);
%\foreach \x in {0,1,...,\m} {
%	\foreach \y in {0,1,...,\n} {
%		\fill[color=gray] (\x,\y) circle (0.1);
%        }
%    }
%\foreach \x in {1,2,...,\m} {
%	\foreach \y in {1,2,...,\n} {
%		\pgfmathsetmacro\c{int(\M*(\y) - \N*(\x-1)+\i)}
%		\node at (\x-0.5, \y-0.5) {\small$\c$};
%        }
%    }
%\draw[very thick, red] (0,0) -- (0,1) -- (1,1) -- (1,2) -- (3,2);
%\end{tikzpicture}
%\hspace{5pt}
%\begin{tikzpicture}[scale=0.5]
%\def\M{6}
%\def\N{4}
%\def\m{3}
%\def\n{2}
%\def\i{1}
%\filldraw[yellow!30!white] (-3,0) rectangle (0,1);
%\filldraw[yellow!30!white] (0,1) rectangle (1,2);
%\draw[gray, <->] (-4,0) -- (2,0);
%\draw[gray, <->] (0,-1) -- (0,\n+1);
%\foreach \x in {-3,...,2} {
%	\foreach \y in {0,1,...,\n} {
%		\fill[color=gray] (\x,\y) circle (0.1);
%        }
%    }
%\foreach \x in {-2,...,2} {
%	\foreach \y in {1,2,...,\n} {
%		\pgfmathsetmacro\c{int(\M*(\y) - \N*(\x-1)+\i)}
%		\node at (\x-0.5, \y-0.5) {\small$\c$};
%        }
%    }
%\draw[very thick, red] (-3,0) -- (-3,1) -- (0,1) -- (0,2);
%
%
%\end{tikzpicture}
%
%\end{center}
%\caption{Here are two path tuples (one in blue and one in red) for $M = 6$, $N = 4$, $k = 2$, $m = \frac{6}{2} = 3$, $n = \frac{4}{2} = 2$. The blue path tuple has two forced north steps (6 and 19) while the red path tuple has just one (6).  Cells that contribute to the area are shaded yellow. Both path tuples have 8 path diagonal inversions.}
%\label{fig:tuples}
%\end{figure}



\subsection{Carlsson--Mellit operators and characteristic functions}
\label{ssec:chi}

Next, we use certain operators defined by Carlsson and Mellit to assign a characteristic function to each path tuple. 

\begin{definition}
For any integer $\ell \geq 0$, we let
$$V_{\ell} = \mathbb{Q}[y_1, y_2, \ldots, y_{\ell}] \otimes \Lambda.$$
Following Carlsson and Mellit \cite{carlsson-mellit}, we define operators
\begin{align*}
d_{+} &\colon V_{\ell} \to V_{\ell + 1} \qquad (\ell \geq 0), \\
d_{-} &\colon V_{\ell} \to V_{\ell - 1} \qquad (\ell \geq 1),
\end{align*} by 
\begin{align*}
d_+ f &= T_1 T_2 \ldots T_{\ell} f[X + (t-1)y_{\ell+1}], \\
d_- f &= \left. -y_{\ell} f[X - (t-1)y_{\ell}] \sum_{i \geq 0} h_i[-X/y_{\ell}] \right|_{y_{\ell}^0},
\end{align*}
where
$$T_i f = \frac{(t-1) y_i f + (y_{i+1} - ty_i) s_i f}{y_{i+1} - y_i}$$
and $s_i$ swaps $y_i$ and $y_{i+1}$.
We define a third operator, 
$$d_{=} \colon V_{\ell} \to V_{\ell} \qquad (\ell \geq 1)$$
which also appears in Carlsson and Mellit's work but not by this name. It acts by 
$$d_{=} f = \frac{1}{t-1} \left(d_- d_+ f - d_+ d_- f \right).$$ 
\end{definition}

Given a path tuple $\bm{P}$, we describe how to obtain a sequence of $d_{+}$,  $d_{-}$, and $d_=$ operators that, when applied to 1, allow us to define $L(v,w)$.

\begin{definition}
A \emph{partial Schr\"oder path} is a lattice path from $(0,\ell)$ to $(N,N$) consisting of steps $(1,0)$, $(0,1)$ and $(1,1)$ that remains weakly above the line $y = x$ and does not contain any diagonal steps on the line $y=x$.
\end{definition}

\begin{definition}
\label{def:schroeder}
Given a path tuple $\bm{P}$, suppose $c_1 < c_2 < \ldots < c_N$ are the cells containing north steps in $\bm{P}$ and that $c_1, \ldots, c_{\ell}$ are the basement steps. Place the values $c_1, c_2, \ldots, c_N$ in the first $N$ cells on the line $y=x$ from bottom left to top right. Let $S$ be the (unique) partial Schr\"oder path from $(0,\ell)$ to $(N, N)$ such that 
\begin{enumerate}[label={(\roman*)}]
\item if $c_i < c_j < c_i + M$, then the entire (unique) square above $c_i$ and to the left of $c_j$ is under $S$,
\item if $c_i + M = c_j$, then the square above $c_i$ and to the left of $c_j$ contains a diagonal step,
\item otherwise, the square above $c_i$ and to the left of $c_j$ is above $S$.
\end{enumerate}
We call $S$ the \emph{partial Schr\"oder path} of $\bm{P}$. 
\end{definition} 

%\begin{figure}
%\begin{center}
%\begin{tikzpicture}[scale=0.5]
%\draw[gray, very thin] (0,0) grid (4,4);
%
%\node at (0.5,0.5) {\small 6};
%\node at (1.5,1.5) {\small 12};
%\node at (2.5,2.5) {\small 19};
%\node at (3.5,3.5) {\small 21};
%
%\draw[very thick, blue] (0,0) -- (0,1) -- (1,2) -- (2,2) -- (2,4) -- (4,4);
%
%\end{tikzpicture}
%\hspace{50pt}
%\begin{tikzpicture}[scale=0.5]
%\draw[gray, very thin] (0,0) grid (4,4);
%
%\node at (0.5,0.5) {\small 6};
%\node at (1.5,1.5) {\small 8};
%\node at (2.5,2.5) {\small 13};
%\node at (3.5,3.5) {\small 19};
%
%\draw[very thick, red] (0,0) -- (0,2) -- (1,2) -- (1,3) -- (2,3) -- (3,4) -- (4,4);
%
%\end{tikzpicture}
%\end{center}
%\caption{This figure contains the Schr\"oder paths of the two mazes from Figure \ref{fig:mazes}. Their corresponding characteristic functions are $d_- d_= d_+ d_- d_- d_+ d_+ (1)$ and $d_- d_- d_+ d_- d_+ d_= d_+(1)$.}
%\label{fig:schroeder}
%\end{figure}

\begin{definition}
Given the partial Schr\"oder path $S$ of a path tuple $\bm{P}$, we begin with 1 and, reading $S$ from right to left, iteratively apply
\begin{itemize}
\item $d_+$ for any horizontal step,
\item $d_-$ for any vertical step, and 
\item $d_=$ for any diagonal step.
\end{itemize}
The resulting element of $V_{\ell}$ is the \emph{characteristic function} of $\bm{P}$, written $\chi(\bm{P})$. 
\end{definition}

The partial Schr\"oder paths and characteristic functions for the path tuples in Figure \ref{fig:paths} are depicted in Figure \ref{fig:schroeder}.
We close this subsection by defining our main objects of study.

\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=0.5]
\draw[gray, very thin] (0,0) grid (4,4);

\node at (0.5,0.5) {\small 3};
\node at (1.5,1.5) {\small 4};
\node at (2.5,2.5) {\small 6};
\node at (3.5,3.5) {\small 9};

\draw[very thick, blue] (0,2) -- (0,3) -- (1,4) -- (4,4);

\end{tikzpicture}
\hspace{50pt}
\begin{tikzpicture}[scale=0.5]
\draw[gray, very thin] (0,0) grid (4,4);

\node at (0.5,0.5) {\small 3};
\node at (1.5,1.5) {\small 4};
\node at (2.5,2.5) {\small 9};
\node at (3.5,3.5) {\small 10};

\draw[very thick, red] (0,2) -- (2,4) -- (4,4);

\end{tikzpicture}
\end{center}
\caption{This figure contains the partial Schr\"oder paths of the two path tuples from Figure \ref{fig:paths}. Their corresponding characteristic functions are $d_- d_= d_+ d_+ d_+(1)$ and $d_= d_= d_+ d_+ (1)$, both of which are in $V_2$.}
\label{fig:schroeder}
\end{figure}


\begin{definition}
Given sequences $v$ and $w$ of lengths $M$ and $N$, respectively, and $|v| = |w|$, we let
$$L(v, w)= \sum_{\bm{P}} t^{-\pdinv(\bm{P})} q^{\area(\bm{P})} \chi(\bm{P}) $$
where the sum is over all $v,w$-path tuples $\bm{P}$. 
\end{definition}

\begin{example}
To complete this subsection, we compute $L(v, w)$ in its entirety for $v = 000110$ and $w=0110$. There are two path tuples in this case, which appear in Figure \ref{fig:paths}. As mentioned in Figure \ref{fig:paths}, the areas of the path tuples are 0 and 1, respectively, and both tuples have 4 path diagonal inversions. Figure \ref{fig:schroeder} depicts the partial Schr\"oder paths for these path tuples. Assembling this information, and evaluating the relevant sequences of Carlsson--Mellit operators, we get
$$
L(000110, 0110) = t^{-4} d_- d_= d_+ d_+ d_+(1) + t^{-4} q d_= d_= d_+ d_+ (1)  = -t^{-2} y_1 s_1 + t^{-3} q y_1 y_2 \in V_2.
$$
\end{example}

\subsection{Compressed path tuples}
\label{ssec:compressed}

By modifying the underlying objects, we can write $L(v, w)$ as a sum over a finite collection of path tuples instead of a potentially infinite collection. 


\begin{definition}
Given sequences $v$ and $w$, a path tuple $\bm{P} = (P^{(0)}, P^{(1)}, \ldots, P^{(k-1)})$ is \emph{compressed} if any $c > 0$ such that
\begin{itemize}
\item all of the cells $c-1, c-2, \ldots, c-M$ are below the path, and 
\item none of the cells $c-1, c-2, \ldots, c-M$ have a north step
\end{itemize}
has a north step. We call any north step in such a cell $c$ a \emph{forced} north step and denote the number of forced north steps in $\bm{P}$ by $\force(\bm{P})$. 
\end{definition}

Then we have the following finite formula for $L(v, w)$.

\begin{thm}
$$L(v, w) = \sum_{\bm{P}} t^{-\pdinv(\bm{P})} q^{\area(\bm{P})} (1-q)^{-\force(\bm{P})} \chi(\bm{P}),$$
where the sum is over all compressed path tuples $\bm{P}$.
\end{thm}

Figure \ref{fig:compressed} shows all compressed paths for $v = 0000$ and $w = 000$.



\subsection{Path labelings when $v=0^M$ and $w = 0^N$}

We can simplify the previous definitions for $L(v, w)$ when $v = 0^M$ and $w = 0^N$. As a result, we get an expression for $L(v,w)$ that does not explicitly use Carlsson--Mellit operators. We can assign labels to north steps in our paths in a manner reminiscent of parking functions. 

\begin{definition}
A \emph{labeled path tuple} is a $k$-tuple of paths $\bm{P}$ along with a function $f$ from the north steps of $\bm{P}$ to the positive integers such that, if $c + M = d$, then $f(c) < f(d)$. 
\end{definition}

We define a notion of diagonal inversions for labeled path tuples.

\begin{definition}
Given a labeled path tuple $(\bm{P}, f)$, a \emph{labeled diagonal inversion} is a pair of cells $c$ and $d$ with north steps such that $c < d < c + M$ and $f(c) < f(d)$. (Again, $c$ and $d$ may be in different sheets.) We let $\ldinv(\bm{P}, f)$ be the number of labeled diagonal inversions in the labeled path tuple $(\bm{P}, f)$.
\end{definition}


\begin{definition}
We let
$$L_{M,N} = \sum_{(\bm{P}, f)} t^{\ldinv(\bm{P}, f) - \pdinv(\bm{P})} q^{\area(\bm{P})} (1-q)^{-\force(\bm{P})}  \prod_{i > 0} x_i^{f^{-1}(i)}, $$
where the sum is over all labeled path tuples $(\bm{P}, f)$ for $M$ and $N$ where $\bm{P}$ is compressed. \end{definition}

As an example, we show all compressed path tuples that contribute to $L_{4,3}$ in Figure \ref{fig:compressed}. The following theorem comes directly from the work of Carlsson and Mellit \cite{carlsson-mellit}.

\begin{thm}
\label{thm:L_M_N}
For positive integers $M$ and $N$, 
$$L_{M,N} = L(0^M, 0^N).$$
\end{thm}


\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=0.35]
\draw[gray, <->] (-5,0) -- (2,0);
\draw[gray, <->] (0,-1) -- (0,4);
\foreach \x in {-5,...,1} {
	\foreach \y in {0,...,3} {
		\fill[color=gray] (\x,\y) circle (0.1);
        }
    }
\foreach \x in {-4,...,1} {
	\foreach \y in {1,...,3} {
		\pgfmathsetmacro\c{int(4*(\y-1) - 3*(\x))}
		\node at (\x-0.5, \y-0.5) {\small$\c$};
        }
    }
\draw[very thick, blue] (-3,0) -- (-3,1) -- (-1,1) -- (-1,2) -- (0,2) -- (0,3) -- (1,3);
\end{tikzpicture}
\hspace{5pt}
\begin{tikzpicture}[scale=0.35]
\filldraw[yellow!30!white] (-1,2) rectangle (0,3);
\draw[gray, <->] (-5,0) -- (2,0);
\draw[gray, <->] (0,-1) -- (0,4);
\foreach \x in {-5,...,1} {
	\foreach \y in {0,...,3} {
		\fill[color=gray] (\x,\y) circle (0.1);
        }
    }
\foreach \x in {-4,...,1} {
	\foreach \y in {1,...,3} {
		\pgfmathsetmacro\c{int(4*(\y-1) - 3*(\x))}
		\node at (\x-0.5, \y-0.5) {\small$\c$};
        }
    }
\draw[very thick, brown] (-3,0) -- (-3,1) -- (-1,1) -- (-1,3) -- (1,3);
\end{tikzpicture}
\hspace{5pt}
\begin{tikzpicture}[scale=0.35]
\filldraw[yellow!30!white] (-4,0) rectangle (-3,1);
\draw[gray, <->] (-5,0) -- (2,0);
\draw[gray, <->] (0,-1) -- (0,4);
\foreach \x in {-5,...,1} {
	\foreach \y in {0,...,3} {
		\fill[color=gray] (\x,\y) circle (0.1);
        }
    }
\foreach \x in {-4,...,1} {
	\foreach \y in {1,...,3} {
		\pgfmathsetmacro\c{int(4*(\y-1) - 3*(\x))}
		\node at (\x-0.5, \y-0.5) {\small$\c$};
        }
    }
\draw[very thick, orange] (-4,0) -- (-4,1) -- (-1,1) -- (-1,2) -- (0,2) -- (0,3);
\end{tikzpicture}
\hspace{5pt}
\begin{tikzpicture}[scale=0.35]
\filldraw[yellow!30!white] (-4,0) rectangle (-3,1);
\filldraw[yellow!30!white] (-1,2) rectangle (0,3);
\draw[gray, <->] (-5,0) -- (2,0);
\draw[gray, <->] (0,-1) -- (0,4);
\foreach \x in {-5,...,1} {
	\foreach \y in {0,...,3} {
		\fill[color=gray] (\x,\y) circle (0.1);
        }
    }
\foreach \x in {-4,...,1} {
	\foreach \y in {1,...,3} {
		\pgfmathsetmacro\c{int(4*(\y-1) - 3*(\x))}
		\node at (\x-0.5, \y-0.5) {\small$\c$};
        }
    }
\draw[very thick, green] (-4,0) -- (-4,1) -- (-1,1) -- (-1,3) -- (0,3);
\end{tikzpicture}
\hspace{5pt}
\begin{tikzpicture}[scale=0.35]
\filldraw[yellow!30!white] (-5,0) rectangle (-3,1);
\filldraw[yellow!30!white] (-1,2) rectangle (0,3);
\draw[gray, <->] (-5,0) -- (2,0);
\draw[gray, <->] (0,-1) -- (0,4);
\foreach \x in {-5,...,1} {
	\foreach \y in {0,...,3} {
		\fill[color=gray] (\x,\y) circle (0.1);
        }
    }
\foreach \x in {-4,...,1} {
	\foreach \y in {1,...,3} {
		\pgfmathsetmacro\c{int(4*(\y-1) - 3*(\x))}
		\node at (\x-0.5, \y-0.5) {\small$\c$};
        }
    }
\draw[very thick, purple] (-5,0) -- (-5,1) -- (-1,1) -- (-1,3);
\end{tikzpicture}
\caption{Above, we have drawn the 5 compressed path tuples for $v = 0000$ and $w = 000$. Since $\gcd(M,N) = 1$, each has only one sheet. Cells that contribute to the area are shaded yellow. Each path tuple has 3 path diagonal inversions and 1 forced north step (at cell 4). The usual 5 Dyck paths for this case can be recovered by starting at the lattice point $(-1,1)$ and following the infinite repetition of each path for a total of 3 north steps and 3 east steps.}
\label{fig:compressed}
\end{center}
\end{figure}



While it is theoretically possible to unwind the definitions of the Carlsson--Mellit operators to give a label-based expression for $L(v, w)$ for any $v$ and $w$, the resulting expression is quite technical and not helpful in what we aim to achieve here. Understanding this expression may be a worthwhile endeavor in the future. We close this section by stating our main conjecture.

\begin{conjecture}
\label{conj:main}
For positive integers $M$ and $N$, let $k = \gcd(M, N)$, $m = M/k$, $n=N/k$. Then
$$\mathbf{Q}_{m,n}^{k}(1) = (-1)^{k(n+1)} (1-q)^k t^{C} L_{M,N},$$
where $C$ is the maximum of $\pdinv(\bm{P})$ over all $M,N$-path tuples $\bm{P}$. 
\end{conjecture}

\noindent
We currently do not know of a simple way to compute the value $C$ in Conjecture \ref{conj:main}.



\section{Connections}
\label{sec:connections}

\subsection{The Rational Shuffle Theorem}

When $M$ and $N$ are coprime (so $M=m$, $N=n$, and $k=1$), the Rational Shuffle Theorem gives a combinatorial expression for $\mathbf{Q}_{m,n} (1)$.  We check that our formulas agree. Since $k=1$, there is only one sheet in each path tuple. We obtain the usual depiction of rational parking functions by viewing each of our path tuples in the horizontal band $1 < y \leq n+1$. We can do this same operation for compressed path tuples. For comparison, we shift each of the path tuples in Figure~\ref{fig:compressed} and depict the resulting rational Dyck paths in Figure~\ref{fig:L_example}. Our statistics for $L_{m,n}$ match up with the statistics that appear on the combinatorial side of the Rational Shuffle Theorem.

\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=0.35]
%\draw[gray, <->] (-3,0) -- (4,0);
%\draw[gray, <->] (0,-1) -- (0,4);
\foreach \x in {-3,...,3} {
	\foreach \y in {1,...,4} {
		\fill[color=gray] (\x,\y) circle (0.1);
        }
    }
\foreach \x in {-2,...,3} {
	\foreach \y in {2,...,4} {
		\pgfmathsetmacro\c{int(4*(\y-1) - 3*(\x))}
		\node at (\x-0.5, \y-0.5) {\small$\c$};
        }
    }
\draw[very thick, blue] (-1,1) -- (-1,2) -- (0,2) -- (0,3) -- (1,3) -- (1,4) -- (3,4);
\end{tikzpicture}
\hspace{5pt}
\begin{tikzpicture}[scale=0.35]
\filldraw[yellow!30!white] (-1,2) rectangle (0,3);

\foreach \x in {-3,...,3} {
	\foreach \y in {1,...,4} {
		\fill[color=gray] (\x,\y) circle (0.1);
        }
    }
\foreach \x in {-2,...,3} {
	\foreach \y in {2,...,4} {
		\pgfmathsetmacro\c{int(4*(\y-1) - 3*(\x))}
		\node at (\x-0.5, \y-0.5) {\small$\c$};
        }
    }
\draw[very thick, brown] (-1,1) -- (-1,3) -- (1,3) -- (1,4) -- (3,4);
\end{tikzpicture}
\hspace{5pt}
\begin{tikzpicture}[scale=0.35]
\filldraw[yellow!30!white] (0,3) rectangle (1,4);

\foreach \x in {-3,...,3} {
	\foreach \y in {1,...,4} {
		\fill[color=gray] (\x,\y) circle (0.1);
        }
    }
\foreach \x in {-2,...,3} {
	\foreach \y in {2,...,4} {
		\pgfmathsetmacro\c{int(4*(\y-1) - 3*(\x))}
		\node at (\x-0.5, \y-0.5) {\small$\c$};
        }
    }
\draw[very thick, orange] (-1,1) -- (-1,2) -- (0,2) -- (0,4) -- (3,4);
\end{tikzpicture}
\hspace{5pt}
\begin{tikzpicture}[scale=0.35]
\filldraw[yellow!30!white] (-1,2) rectangle (0,3);
\filldraw[yellow!30!white] (0,3) rectangle (1,4);
\foreach \x in {-3,...,3} {
	\foreach \y in {1,...,4} {
		\fill[color=gray] (\x,\y) circle (0.1);
        }
    }
\foreach \x in {-2,...,3} {
	\foreach \y in {2,...,4} {
		\pgfmathsetmacro\c{int(4*(\y-1) - 3*(\x))}
		\node at (\x-0.5, \y-0.5) {\small$\c$};
        }
    }
\draw[very thick, green] (-1,1) -- (-1,3) -- (0,3) -- (0,4) -- (3,4);
\end{tikzpicture}
\hspace{5pt}
\begin{tikzpicture}[scale=0.35]
\filldraw[yellow!30!white] (-1,3) rectangle (1,4);
\filldraw[yellow!30!white] (-1,2) rectangle (0,3);
\foreach \x in {-3,...,3} {
	\foreach \y in {1,...,4} {
		\fill[color=gray] (\x,\y) circle (0.1);
        }
    }
\foreach \x in {-2,...,3} {
	\foreach \y in {2,...,4} {
		\pgfmathsetmacro\c{int(4*(\y-1) - 3*(\x))}
		\node at (\x-0.5, \y-0.5) {\small$\c$};
        }
    }
\draw[very thick, purple] (-1,1) -- (-1,4) -- (3,4);
\end{tikzpicture}
\caption{We have shifted the 5 compressed path tuples for $v=0000$ and $w=000$ up to obtain rational Dyck paths.}
\label{fig:L_example}
\end{center}
\end{figure}


\subsection{Torus link homology}

For positive integers $M$ and $N$, Hogancamp and Mellit derived a recursion for computing the triply graded Khovanov--Rozansky homology of the $M,N$-torus link~\cite{hogancamp-mellit}. We follow Gorsky, Mazin, and Vazirani's approach by allowing $\bullet$'s to appear in $v$ and $w$~\cite{gorsky-mazin-vazirani}; Hogancamp and Mellit's original recursion can be obtained by removing all $\bullet$'s. We state this theorem below for comparison with our generalization (Theorem \ref{thm:L_recursion}). 

\begin{thm}[\cite{gorsky-mazin-vazirani, hogancamp-mellit}]
\label{thm:p_recursion}
For nonnegative integers $M$ and $N$, the triply graded Khovanov--Rozansky homology of the $M,N$-torus link is free over $\mathbb{Z}$ of graded rank $p(0^M, 0^N)$, which is an element of $\mathbb{N}[q,t^{\pm 1}, a, (1-q)^{-1}]$ computed by the following recursion:
\begin{enumerate}
\setcounter{enumi}{-1}
\item $p(\bullet^M, \bullet^N) = 1$.
\item $p(\bullet v, \bullet w) = p(v \bullet, w \bullet)$.
\item $p(0v, 0w) = (1-q)^{-1} p(v1, w1)$ if $|v| = |w| = 0$.
\item $p(0v, 0w) = t^{-\ell}p(v1, w1) + qt^{-\ell}p(v0, w0)$ if $\ell = |v| = |w| > 0$.
\item $p(1v, 0w) = p(v1, w\bullet)$.
\item $p(0v, 1w) = p(v\bullet, w1)$.
\item $p(1v, 1w) = (t^{|v|} + a) p(v\bullet, w\bullet)$.
\end{enumerate}
\end{thm}

%example

An analogous recursion holds for $L(v, w)$. In fact, the search for such a recursion motivated the discovery of $L(v, w)$. %say more?


\begin{thm}
\label{thm:L_recursion}
Let $v$ and $w$ be sequences in the alphabet $\{0,1,\bullet\}$ with $|v| = |w|$. We can compute $L(v, w)$ via the following recursion:
\begin{enumerate}
\setcounter{enumi}{-1}
\item \label{initial} $L(\bullet^M, \bullet^N) = 1$.
\item \label{bullets} $L(\bullet v, \bullet w) = L(v \bullet, w \bullet)$.
\item \label{all0} $L(0v, 0w) = (1-q)^{-1} d_- L(v1, w1)$ if $|v| = |w| = 0$.
\item \label{00} $L(0v, 0w) = t^{-\ell}d_{-} L(v1, w1) + qt^{-\ell}L(v0, w0)$ if $\ell = |v| = |w| > 0$.
\item \label{10}  $L(1v, 0w) = t^{-\ell} d_= L(v1, w\bullet)$ if $\ell = |v| = |w|$.
\item \label{01} $L(0v, 1w) = L(v\bullet, w1)$.
\item \label{11} $L(1v, 1w) = d_+ L(v\bullet, w\bullet)$
\end{enumerate}

\end{thm}

The proof of Theorem \ref{thm:L_recursion} (which we omit in this extended abstract) is obtained by considering the different ways that cells 0, $M$, and $N$ can appear with respect to a path tuple. Figure~\ref{fig:recursion_proof} contains a visual guide to these different configurations and how they correspond to cases in the recursion. 

One can view Theorem \ref{thm:L_recursion} as a lift of Theorem \ref{thm:p_recursion} to the level of symmetric functions (or, more precisely, symmetric functions tensored with polynomials in variables $y_1, y_2, \ldots, y_{\ell}$). A reasonable approach to proving Conjecture \ref{conj:main} would be to mimic this recursion to define and study extensions of the operators $\mathbf{Q}_{m,n}$ to sequences $v$ and $w$. A similar approach was used by Mellit to prove the Rational Shuffle Theorem~\cite{mellit-rational}. 

It would also be interesting to investigate whether $L(v, w)$ has a precise geometric or topological meaning in particular cases. For example, Hogancamp and Mellit show that $p(1^k 0^{m(k-1)}, 1^{k} 0^{n(k-1)})$ is related to colored homology of certain torus links~\cite{hogancamp-mellit}. We close this abstract by describing how to recover any $p(v,w)$ from the corresponding $L(v,w)$. 



%\begin{proof}
%\eqref{initial} holds since there is exactly one such path, and it does not have any area or path diagonal inversions.
%
%For the other statements, suppose $\bm{P}$ is a path tuple for sequences $iv$ and $jw$ for some symbols $i$ and $j$. $i$ and $j$ describe where the cell 0 is located with respect to the north and east steps of $\bm{P}$, respectively. Let $\bm{P}'$ be the path tuple that is obtained by replacing each content label $c$ in $\bm{P}$ with $c-1$. We will show that $\bm{P}'$ is always a $vi', wj'$-path tuple for some symbols $i'$ and $j'$. Furthermore, since cells $M$ and $N$ appear directly above and to the left of cell 0, respectively, we can use $i$ and $j$ to relate the contribution of $\bm{P'}$ to $L(vi', wj')$ to the contribution of $\bm{P}$ to $L(iv, jw)$. 
%
%In \eqref{bullets}, cell 0 is left and above the path in $\bm{P}$ in sheet 0, so cells $M$ and $N$ must also be. Therefore cells $M-1$ and $N-1$ in $\bm{P}'$ are left and above $\bm{P}$ and the contribution of $\bm{P}'$ matches that of $\bm{P}$.
%
%In \eqref{all0}, $0,1,\ldots,M-1$ are all right of $\bm{P}$ and $0,1,\ldots,N-1$ are all below $\bm{P}$. Let $r \geq 0$ be the smallest cell such that $r+M$ has a north step and $r+N$ has an east step. Then we can decrement the contents of every cell $r$ times to get a $v1, w1$-path tuple. Each time we decrement we lose a cell that contributes to area, which accounts for the $(1-q)^{-1}$ factor.
%
%%In \eqref{all0}, $\bm{P}$ must be an $M,N$-path tuple. Therefore $\bm{P}$ contains a north step at $M$ and an east step at $N$. As a result, $\bm{P}'$ has a north stepath tuplep at $M-1$ and an east step at $N-1$. The north step at $M$ is forced in $\bm{P}$ but not in $\bm{P}'$, which accounts for the $(1-q)^{-1}$ factor. Furthermore, the partial Schr\"oder path of $\bm{P}'$ is identical to that of $\bm{P}$ except that it stops at $(0,1)$ instead of taking the final $d_-$ step down to $(0,0)$. 
%
%In \eqref{00}, 0 is to the right of a north step and below an east step, but it is not directly adjacent to either step. The ``corner'' (east step followed by a north step) on the horizontal line on 0's north edge either appears at the upper left vertex of the 0 cell or to the left of this vertex. If it appears at the upper left vertex, then $M$ contains a north step and $N$ contains an east step. These steps become basement steps at $M-1$ and $N-1$, respectively, in $\bm{P}'$, and we get a $d_-$ in the first summand on the right-hand side of \eqref{00}. If the corner above 0 does not occur at the upper left vertex of cell 0, we do not gain any new basement steps, so $i=j=0$ and we do not see a new $d_-$. Furthermore, cell $M+N$ counts toward the area of $\bm{P}$ but not that of $\bm{P}'$. Finally, in either case, there are $\ell$ path diagonal inversions of the form $(0, c)$ in $\bm{P}$ that are not path diagonal inversions in $\bm{P}'$. 
%
%In \eqref{10}, 0 has a north step but not an east step. Therefore $M$ has a north step and $N$ is above an east step in $\bm{P}$. In $\bm{P}'$, $M-1$ has a north step and $N-1$ is above an east step. Furthermore, there are $\ell$ path diagonal inversions of the form $(c,M)$ in $\bm{P}$ that are not path diagonal inversions in $\bm{P}'$. Finally, the partial Schr\"oder path of $\bm{P}$ has a diagonal step formed by 0 and $M$ that does not exist in the partial Schr\"oder path of $\bm{P}'$. 
%
%In \eqref{01}, 0 has an east step but not a north step. Similarly, $M-1$ is to the left of a north step and $N-1$ has an east step in $\bm{P}'$. The contributions of $\bm{P}$ and $\bm{P}'$ remain the same. 
%
%Finally, in \eqref{11}, 0 has a north step and an east step, which means that, in $\bm{P}'$, $M-1$ is to the left of a north step and $N-1$ is above an east step. 0 appears at the bottom of the partial Schr\"oder path for $\bm{P}$ but $-1$ cannot occur in the partial Schr\"oder path of $\bm{P}'$, which accounts for the $d_+$ in \eqref{11}.
%
%
%
%
%\end{proof}

\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=0.5]
\node at (0.5, 0.5) {\small$0$};
\node at (0.5, 1.5) {\small$M$};
\node at (-0.5, 0.5) {\small$N$};
\draw[very thick, dashed] (-1,0) -- (2,0) -- (2,2);
\node at (0.5,-2) {\eqref{bullets}};
\end{tikzpicture}
\hspace{20pt}
\begin{tikzpicture}[scale=0.5]
\node at (0.5,-2) {\eqref{all0}};
\node at (0.5, 0.5) {\small$0$};
\node at (0.5, 1.5) {\small$M$};
\node at (-0.5, 0.5) {\small$N$};
\draw[very thick, dashed] (-1,1) -- (0,1) -- (0,2);
\end{tikzpicture}
\hspace{20pt}
\begin{tikzpicture}[scale=0.5]
\node at (2,-2) {\eqref{00}};
\node at (0.5, 0.5) {\small$0$};
\node at (0.5, 1.5) {\small$M$};
\node at (-0.5, 0.5) {\small$N$};
\draw[very thick] (-1,1) -- (0,1) -- (0,2);
\node at (4.5, 0.5) {\small$0$};
\node at (4.5, 1.5) {\small$M$};
\node at (3.5, 0.5) {\small$N$};
\draw[very thick] (2,1) -- (3,1) -- (3,2);
\end{tikzpicture}
\hspace{20pt}
\begin{tikzpicture}[scale=0.5]
\node at (0.5,-2) {\eqref{10}};
\node at (0.5, 0.5) {\small$0$};
\node at (0.5, 1.5) {\small$M$};
\node at (-0.5, 0.5) {\small$N$};
\draw[very thick, dashed] (0,0) -- (0,1);
\draw[very thick] (0,1) -- (0,2);
\end{tikzpicture}
\hspace{20pt}
\begin{tikzpicture}[scale=0.5]
\node at (0.5,-2) {\eqref{01}};
\node at (0.5, 0.5) {\small$0$};
\node at (0.5, 1.5) {\small$M$};
\node at (-0.5, 0.5) {\small$N$};
\draw[very thick] (-1,1) -- (0,1);
\draw[very thick, dashed] (0,1) -- (1,1);
\end{tikzpicture}
\hspace{20pt}
\begin{tikzpicture}[scale=0.5]
\node at (0.5,-2) {\eqref{11}};
\node at (0.5, 0.5) {\small$0$};
\node at (0.5, 1.5) {\small$M$};
\node at (-0.5, 0.5) {\small$N$};
\draw[very thick, dashed] (0,0) -- (0,1) -- (1,1);
\end{tikzpicture}
\end{center}
\caption{We sketch the different possible configurations of steps near cells 0, $M$, and $N$ in a path tuple $\bm{P}$ which correspond to the different recursive cases in Theorem \ref{thm:L_recursion}.}
\label{fig:recursion_proof}
\end{figure}

%example


\begin{corollary}
Let $\psi$ be the operator\footnote{To experts in this area, applying $\psi$ equivalent to taking the ``Schr\"oder inner product.''} on $\Lambda$ defined by $\psi(e_i) = 1+a$ for each (algebraically independent) $e_i$. Then, for any sequences $v$ and $w$ in the alphabet $\{0,1,\bullet\}$ with $|v| = |w| = \ell$, 
$$ \psi\left( d_-^{\ell} L(v, w) \right) = p(v, w).$$
\end{corollary}




\acknowledgements{}
The author would like to thank AIM's Link Homology Research Community for the opportunity to give a talk on the $m=n$ case of this work. The audience's feedback from that talk inspired much of this research. The author would also like to thank the referees for their helpful comments.


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