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\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}

\usepackage{lipsum}

%% define your title in the usual way
\title{A conjectural basis for the \texorpdfstring{$(1,2)$}{(1,2)}-bosonic-fermionic coinvariant ring}


%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author[J. Lentfer]{\texorpdfstring{John Lentfer\thanks{\href{mailto:jlentfer@berkeley.edu}{jlentfer@berkeley.edu}. The author was partially supported by the National Science Foundation Graduate Research Fellowship DGE-2146752. This is an extended abstract of \cite{Lentfer2024}.}\addressmark{1}}{John Lentfer}}


%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of California, Berkeley, CA, USA}

%% put the date of submission here
\received{November 12, 2024}

%% 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{
We give the first conjectural construction of a monomial basis for the coinvariant ring $R_n^{(1,2)}$, for the symmetric group $\mathfrak{S}_n$ acting on one set of bosonic (commuting) and two sets of fermionic (anticommuting) variables. 
Our construction interpolates between the modified Motzkin path basis for $R_n^{(0,2)}$ of Kim--Rhoades (2022) and the super-Artin basis for $R_n^{(1,1)}$ conjectured by Sagan--Swanson (2024) and proven by Angarone et al. (2024). 
We prove that our proposed basis has cardinality $2^{n-1}n!$, aligning with a conjecture of Zabrocki (2020) on the dimension of $R_n^{(1,2)}$, and show how it gives a combinatorial expression for the Hilbert series. 
We also conjecture a Frobenius series for $R_n^{(1,2)}$, including formulas for hook characters and the $m_\mu$ coefficients.
}

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

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Coinvariant ring, Hilbert series, Frobenius series, Artin basis}

%% 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; i.e., [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, giveninits]{biblatex}
\addbibresource{biblio.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

%\tableofcontents

\section{Introduction}\label{sec:introduction}
The classical coinvariant ring $R_n^{(1,0)} = \C[\bm{x}_n]/\allowbreak \langle \C[\bm{x}_n]_+^{\mathfrak{S}_n} \rangle$ is the quotient of a polynomial ring in $n$ variables $\bm{x}_n = \{x_1,\ldots,x_n\}$ by $\mathfrak{S}_n$-invariant polynomials with no constant term.
It is well-known (see for example \cite[Section 1.5]{Haiman1994}) to have dimension $n!$, Hilbert series $[n]_q!$, and Frobenius series \[\sum_{\lambda \vdash n} \sum_{T \in \SYT(\lambda)} q^{\maj(T)}s_\lambda.\]
An important basis of the classical coinvariant ring is the Artin basis.

In 1994, Haiman \cite{Haiman1994} introduced the diagonal coinvariant ring $R_n^{(2,0)} = \C[\bm{x}_n,\bm{y}_n]/\allowbreak \langle \C[\bm{x}_n,\bm{y}_n]_+^{\mathfrak{S}_n} \rangle$, which extends the classical coinvariant ring to two sets of $n$ variables. 
Here and throughout, $\mathfrak{S}_n$ acts diagonally by permuting the indices of the variables. 
%Given a polynomial ring $R$, the notation $\langle R_+^{\mathfrak{S}_n} \rangle$ denotes the ideal generated by all polynomials in $R$ which are invariant under the diagonal action of $\mathfrak{S}_n$, with no constant term.
In 2002, Haiman \cite{Haiman2002} proved that $R_n^{(2,0)}$ has dimension $(n+1)^{n-1}$, Hilbert series $\langle \nabla_{q,t} (e_n) , h_1^n\rangle$, and Frobenius series $\nabla_{q,t} (e_n)$, using several deep results in algebraic geometry.
A combinatorial formula for its Hilbert series was conjectured by Haglund and Loehr \cite{HaglundLoehr2005}, and eventually was proven as a consequence of the more general shuffle theorem, conjectured by Haglund, Haiman, Loehr, Remmel, and Ulyanov \cite{HHLRU2005} and proven by Carlsson and Mellit \cite{CarlssonMellit2018}, which gives a combinatorial formula for $\nabla_{q,t} (e_n)$.
A monomial basis was given by Carlsson and Oblomkov \cite{CarlssonOblomkov2018}.

Recently, there has been interest (see \cite{BergeronOPAC, Bergeron2020, BHIR, Zabrocki2020}) in extending the setting to include coinvariant rings with $k$ sets of $n$ commuting variables $\bm{x}_n, \bm{y}_n, \bm{z}_n, \ldots$ and $j$ sets of $n$ anticommuting variables $\bm{\theta}_n, \bm{\xi}_n, \bm{\rho}_n, \ldots$ (which commute with $\bm{x}_n, \bm{y}_n, \bm{z}_n, \ldots$).
We denote this by \[R_n^{(k,j)} = \C[\underbrace{\bm{x}_n, \bm{y}_n, \bm{z}_n, \ldots}_k, \underbrace{\bm{\theta}_n, \bm{\xi}_n, \bm{\rho}_n, \ldots}_j]/\langle \C[\underbrace{\bm{x}_n, \bm{y}_n, \bm{z}_n, \ldots}_k, \underbrace{\bm{\theta}_n, \bm{\xi}_n, \bm{\rho}_n, \ldots}_j]_+^{\mathfrak{S}_n} \rangle.\]
%Commuting variables commute with all variables. Anticommuting variables anticommute with all anticommuting variables, that is, $\theta_i \theta_j = - \theta_j \theta_i$, which also implies that $\theta_i^2 = 0$. 

We briefly overview recent results on some special cases of $R_n^{(k,j)}$ for which there has been significant progress. Kim and Rhoades \cite{KimRhoades2022} showed that the fermionic diagonal coinvariant ring $R_n^{(0,2)} = \C[\bm{\theta}_n,\bm{\xi}_n]/\allowbreak \langle \C[\bm{\theta}_n,\bm{\xi}_n]_+^{\mathfrak{S}_n} \rangle$ has dimension $\binom{2n-1}{n}$, confirming a conjecture of Zabrocki \cite{Zabrocki2020}. They gave a combinatorial formula for its Hilbert series:
\begin{equation}\Hilb(R_n^{(0,2)}; u, v) = \sum_{\pi \in \Pi(n)_{> 0}} u^{\deg_\theta(\pi)} v^{\deg_\xi(\pi)},\end{equation}
where $\Pi(n)_{> 0}$ denotes the set of modified Motzkin paths of length $n$. They also gave a monomial basis for $R_n^{(0,2)}$ and found its Frobenius series \cite[Theorem 6.1]{KimRhoades2022}.
%\begin{equation}\label{eq:02frob} \Frob(R_n^{(0,2)};u,v) = \sum_{i+j < n} s_{(n-i,1^i)}*s_{(n-j,1^j)}-s_{(n-i+1,1^{i-1})}*s_{(n-j+1,1^{j-1})},\end{equation} where $*$ denotes the Kronecker product and $s_{(n+1,1^{-1})}$ is interpreted as $0$.

The superspace coinvariant ring is $R_n^{(1,1)} = \C[\bm{x}_n,\bm{\theta}_n]/\allowbreak \langle \C[\bm{x}_n,\bm{\theta}_n]_+^{\mathfrak{S}_n} \rangle$. 
Sagan and Swanson \cite{SaganSwanson2024} conjectured, and Rhoades and Wilson \cite{RhoadesWilson2023} proved, that its Hilbert series is
\begin{equation} 
    \Hilb(R_n^{(1,1)}; q; u) = \sum_{k=1}^n u^{n-k} [k]_q! \Stir_q(n,k),
\end{equation}
where $\Stir_q(n,k)$ is a $q$-analogue of the Stirling numbers.  
The dimension of $R_n^{(1,1)}$ is the ordered Bell number, which counts the number of ordered set partitions of $\{1,\ldots,n\}$.
Sagan and Swanson \cite{SaganSwanson2024} conjectured and Angarone, Commins, Karn, Murai, and {\parfillskip=0pt \par}
\noindent Rhoades \cite{Angarone2024} proved that a certain super-Artin set is a basis for $R_n^{(1,1)}$.
A Frobenius series has been conjectured for $R_n^{(1,1)}$ \cite[Equation 5.2]{BHIR}.
%\begin{equation}\label{eq:11conjFrob} \Frob(R_n^{(1,1)};q;u) = \sum_{k=0}^{n-1} \sum_{\lambda \vdash n} \sum_{T \in \SYT(\lambda)} u^k q^{\maj(T) - k\des(T) + \binom{k}{2}}  \qbinom{\des(T)}{k}_q s_\lambda.\end{equation}

%Zabrocki \cite{Zabrocki2019} conjectured the following Frobenius series for $R_n^{(2,1)}$: \begin{equation}\label{eq:zabrocki}
%\Frob(R_n^{(2,1)};q,t;u) = \Delta'_{e_{n-1} + u e_{n-2} + \cdots + u^{n-1}}(e_n),
%\end{equation} where $\Delta_f'$ is a Macdonald eigenoperator defined by $\Delta_f'\tilde{H}_\mu[X;q,t] = f[B_\mu(q,t) - 1]\tilde{H}_\mu[X;q,t]$.\footnote{For definitions of the terms in this expression, see \cite{HaglundRemmelWilson2018}.} Finally, its dimension is conjectured by Zabrocki \cite{Zabrocki2020} to be given by $\frac{1}{2}\sum_{k=0}^{n+1} \binom{n+1}{k}\frac{k^n}{n+1}$ (see OEIS sequence \href{https://oeis.org/A201595}{A201595} \cite{OEIS}). Haglund, Rhoades, and Shimozono showed that this implies the conjecture in equation~\eqref{eq:11conjFrob} \cite{HaglundRhoadesShimozono}.
Zabrocki \cite{Zabrocki2019} conjectured a Frobenius series for $R_n^{(2,1)}$.
Then D'Adderio, Iraci, and Vanden Wyngaerd \cite{DadderioIraciVandenWyngaerd2021} introduced certain symmetric function operators $\Theta_f$, called Theta operators, where $f$ is any symmetric function, to extend Zabrocki's conjecture to a conjectural Frobenius series for $R_n^{(2,2)}$. They conjectured that
\begin{equation}\label{eq:thetaconj} \Frob(R_n^{(2,2)}; q,t;u,v) = \sum_{\substack{k,\ell \geq 0,\\ k + \ell < n}} u^k v^\ell\Theta_{e_k}\Theta_{e_\ell}\nabla_{q,t} (e_{n-k-\ell}),\end{equation}
which is known as the Theta conjecture.

The ring $R_n^{(1,2)} = \C[\bm{x}_n,\bm{\theta}_n,\bm{\xi}_n]/\allowbreak \langle \C[\bm{x}_n,\bm{\theta}_n,\bm{\xi}_n]_+^{\mathfrak{S}_n} \rangle$ is the main object of study in this paper. 
Zabrocki \cite{Zabrocki2020} conjectured that the dimension of $R_n^{(1,2)}$ is $2^{n-1}n!$, and an ungraded Frobenius series was conjectured by Bergeron \cite{Bergeron2020}.
%\begin{equation}\label{eq:Bergeron_conj1} \Frob(R_n^{(1,2)};q;u,v)|_{q=u=v=1} = \frac{1}{2} \sum_{\mu \vdash n} 2^{\ell(\mu)}(-1)^{n-\ell(\mu)} \binom{\ell(\mu)}{d_1(\mu),\ldots,d_n(\mu)} p_\mu,\end{equation}
%where $\ell(\mu)$ denotes the length of the partition $\mu$ and $d_i(\mu)$ denotes the number of parts of size $i$ in $\mu$.
The Theta conjecture specialized at $t=0$ immediately gives a conjectural Frobenius series for $R_n^{(1,2)}$, however, the specialization can only be done after applying both of the Theta operators, so the resulting formula does not easily simplify.
Recently, Iraci, Nadeau, and Vanden Wyngaerd \cite{IraciNadeauVandenWyngaerd2024} have given a conjectural Hilbert series and conjectural Frobenius series using the combinatorics of segmented permutations. However, they did not propose a monomial basis.

Our first main contribution is a combinatorial construction of a set of monomials, denoted by $B_n^{(1,2)}$ (\Cref{def:B_n^{(1,2)}}), which we conjecture to give a basis of $R_n^{(1,2)}$ (\Cref{conj:R_n^{(1,2)}}).
If the conjecture holds, it implies the following combinatorial Hilbert series (\Cref{prop:Hilb(R_n^{(1,2)})}):
\begin{equation}\label{eq:comb_Hilbert} \Hilb(R_n^{(1,2)};q;u,v) = \sum_{\pi \in \Pi(n)_{>0}} u^{\deg_\theta(\pi)} v^{\deg_\xi(\pi)} \stair_q(\pi).\end{equation}
In support of this conjecture, we show that the cardinality of $B_n^{(1,2)}$ is $2^{n-1}n!$ (\Cref{thm:cardinality}). 
By establishing a weight-preserving bijection between $B_n^{(1,2)}$ and the set of segmented permutations $\SW(1^n)$ (\Cref{thm:bijection}), it follows that our conjectural Hilbert series is equivalent to a conjectural Hilbert series of Iraci, Nadeau, and Vanden Wyngaerd \cite{IraciNadeauVandenWyngaerd2024}.

Our second main contribution is a simple combinatorial formula for the conjectural Frobenius series of $R_n^{(1,2)}$ (\Cref{conj:frob}):
\begin{equation}\label{eq:conj-frob-intro}\Frob(R_n^{(1,2)};q;u,v) = \sum_{b \in B_n^{(1,2)}} u^{\deg_\theta(b)} v^{\deg_\xi(b)} q^{\deg_x(b)} Q_{\Asc(b),n},\end{equation}
where $Q_{S,n}$ denotes the fundamental quasisymmetric function.
We show that this is equivalent to the conjectural Frobenius series of Iraci, Nadeau, and Vanden Wyngaerd (\Cref{thm:equiv_Frobenius}).
A benefit of using $B_n^{(1,2)}$ instead of segmented permutations is that determining $\Asc(b)$ and $\deg_x(b)$ for $b \in B_n^{(1,2)}$ is typically more direct than determining $\Split(\sigma)$ and $\sminv(\sigma)$ for $\sigma \in \SW(1^n)$, which fulfill analogous roles.

\Cref{sec:everything-else} contains some applications demonstrating the utility of our constructions. We give a formula for the $m_\mu$ coefficient of our conjectural Frobenius series, which can also be interpreted as a symmetric function identity. 
We give a formula for the coefficient of a hook Schur function $s_{(d+1,1^{n-d-1})}$ in the symmetric function in \Cref{eq:conj-frob-intro}: 
\[ \sum_{k+\ell < n} u^k v^\ell q^{\binom{n-d-k-\ell}{2}}\qbinom{n-1-d}{\ell}_q\qbinom{n-1-k}{d}_q\qbinom{n-1-\ell}{k}_q,\]
extending a result of Iraci, Nadeau, and Vanden Wyngaerd on column Schur functions $s_{(1^{n})}$.

Full results with proofs advertised in this extended abstract, along with some extensions to type $B$, can be found in \cite{Lentfer2024}.

\section{Background}\label{sec:background}
In this section, we describe the Kim--Rhoades basis for $R_n^{(0,2)}$ \cite{KimRhoades2022} and the super-Artin basis for $R_n^{(1,1)}$ conjectured by Sagan--Swanson \cite{SaganSwanson2024} and proven by Angarone, Commins, Karn, Murai, and Rhoades \cite{Angarone2024}.

Define the set of \textit{modified Motzkin paths} of length $n$, $\Pi(n)_{>0}$, to be the set of all paths $\pi = (p_1,\ldots,p_n)$ in $\Z^2$ such that each step $p_i$ is one of: 
\begin{enumerate}[label = (\alph*)]
    \item an up-step $(1,1)$,
    \item a horizontal step $(1,0)$ with decoration $\theta_i$,
    \item a horizontal step $(1,0)$ with decoration $\xi_i$,
    \item or a down-step $(1,-1)$ with decoration $\theta_i\xi_i$,
\end{enumerate}   
where the first step must be an up-step, and subsequently, the path never goes below the horizontal line $y=1$.\footnote{Kim and Rhoades defined the modified Motzkin paths in a slightly different manner, where there are no decorations on the down-steps, but they still contribute $\theta_i\xi_i$ to the weight. Furthermore, they defined the decorations to be just $\theta$ or $\xi$ instead of $\theta_i$ and $\xi_i$. Converting between these conventions is straightforward.}

Define the \textit{weight} $\wt(p_i)$ of a step $p_i$ of a modified Motzkin path to be its decoration, or $1$ if it does not have a decoration. Then define the \textit{weight} $\wt(\pi)$ of a modified Motzkin path $\pi \in \Pi(n)_{>0}$ to be the product of the weights of each step $p_i$, that is,
\[ \wt(\pi) := \prod_{p_i \in (p_1,\ldots,p_n) \, =\,  \pi} \wt(p_i).\]

\begin{definition}[\cite{KimRhoades2022}]
    The \textit{Kim--Rhoades basis} $B_n^{(0,2)}$ is the set of all weights of the modified Motzkin paths $\pi \in \Pi(n)_{>0}$, that is,
    \[ B_n^{(0,2)} := \{ \wt(\pi)\, | \, \pi \in \Pi(n)_{>0}\}.\]
\end{definition} 

For an example, see \Cref{fig:Motzkin_3_example}.

\begin{theorem}[\cite{KimRhoades2022}]
    The Kim--Rhoades basis $B_n^{(0,2)}$ is a basis for $R_n^{(0,2)}$.
\end{theorem}

\begin{figure}[!htb]
    \centering
    \begin{tikzpicture}
        \def\sep{2.6}
        \node[anchor=south west] at (0, 0) {
            \motzkin{up,up,up}{1}
        };
        \node[anchor=south west] at (4*\sep, {0*\sep}) {
            \motzkin{up,flat-theta,up}{$\theta_2$}
        };
        \node[anchor=south west] at ({0*\sep}, {-1*\sep}) {
            \motzkin{up,flat-xi,up}{$\xi_2$}
        };
        \node[anchor=south west] at (\sep, {0*\sep}) {
            \motzkin{up,up,flat-theta}{$\theta_3$}
        };
        \node[anchor=south west] at ({2*\sep}, {0*\sep}) {
            \motzkin{up,up,flat-xi}{$\xi_3$}
        };
        \node[anchor=south west] at (1*\sep, {-1*\sep}) {
            \motzkin{up,flat-theta,flat-theta}{$\theta_2\theta_3$}
        };
        \node[anchor=south west] at ({3*\sep}, {0*\sep}) {
            \motzkin{up,up,down}{$\theta_3\xi_3$}
        };
        \node[anchor=south west] at ({3*\sep}, {-1*\sep}) {
            \motzkin{up,flat-xi,flat-theta}{$\xi_2\theta_3$}
        };
        \node[anchor=south west] at ({2*\sep}, {-1*\sep}) {
            \motzkin{up,flat-theta,flat-xi}{$\theta_2\xi_3$}
        };
        \node[anchor=south west] at ({4*\sep}, {-1*\sep}) {
            \motzkin{up,flat-xi,flat-xi}{$\xi_2\xi_3$}
        };
    \end{tikzpicture}
    \caption{The basis $B_3^{(0,2)}$. Each modified Motzkin path is labeled with its corresponding monomial.}
    \label{fig:Motzkin_3_example}
\end{figure}

Next, we recall the super-Artin basis, defined by Sagan and Swanson. Let $\chi(P)$ be $1$ if the proposition $P$ is true, and $0$ if the proposition $P$ is false. Let $\theta_T$ denote the ordered product $\theta_{t_1} \cdots \theta_{t_k}$ for any subset $T = \{ t_1 < \cdots < t_k\} \subseteq \{1,\ldots,n\}$. For any $T \subseteq \{2,\ldots, n\}$, define the \textit{$\alpha$-sequence} $\alpha(T) = (\alpha_1(T), \ldots, \alpha_n(T))$ recursively by the initial condition $\alpha_1(T) = 0$ and for $2 \leq  i \leq n$, 
\[
    \alpha_i(T) =\alpha_{i-1}(T) + \chi(i \not \in T).
\]

\begin{definition}[\cite{SaganSwanson2024}]
    The super-Artin set is \[B_n^{(1,1)} := \{x^\alpha \theta_T \, | \, T \subseteq \{2,\ldots,n\} \text{ and } \alpha \leq \alpha(T) \text{ componentwise}\}.\]
\end{definition}

See \Cref{fig:super_Artin_3_example} for an example. The following result was conjectured by Sagan and Swanson, and was proven by Angarone, Commins, Karn, Murai, and Rhoades.

\begin{theorem}[\cite{Angarone2024}]
    The super-Artin set $B_n^{(1,1)}$ is a basis for $R_n^{(1,1)}$.
\end{theorem}

\begin{figure}[!htb]
    \centering
    \begin{tikzpicture}
        \def\sep{2.6}
        \def\hs{0.92}
        \node[anchor=south west] at (0, 0) {
            \motzkinpath{skip,skip,skip}{0,1,2}{0,0,0}{1}
        };
        \node[anchor=south west] at ({\sep}, 0) {
            \motzkinpath{skip,skip,skip}{0,1,2}{0,0,1}{$x_3$}
        };
        \node[anchor=south west] at ({2*\sep}, 0) {
            \motzkinpath{skip,skip,skip}{0,1,2}{0,1,0}{$x_2$}
        };
        \node[anchor=south west] at ({3*\sep}, 0) {
            \motzkinpath{skip,skip,skip}{0,1,2}{0,1,1}{$x_2x_3$}
        };
        \node[anchor=south west] at ({4*\sep}, 0) {
            \motzkinpath{skip,skip,skip}{0,1,2}{0,0,2}{$x_3^2$}
        };
        \node[anchor=south west] at ({0*\sep}, -1*\sep*\hs) {
            \motzkinpath{skip,skip,skip}{0,1,2}{0,1,2}{$x_2x_3^2$}
        };
        \node[anchor=south west] at (1*\sep, {-2*\sep*\hs*\hs}) {
            \motzkinpath{skip,skip,skip}{0,0,1}{0,0,0}{$\theta_2$}
        };
        \node[anchor=south west] at ({2*\sep}, {-1*\sep*\hs}) {
            \motzkinpath{skip,skip,skip}{0,1,1}{0,0,1}{$x_3\theta_3$}
        };
        \node[anchor=south west] at ({4*\sep}, {-1*\sep*\hs}) {
            \motzkinpath{skip,skip,skip}{0,1,1}{0,1,1}{$x_2x_3\theta_3$}
        };
        \node[anchor=south west] at (\sep, {-1*\sep*\hs}) {
            \motzkinpath{skip,skip,skip}{0,1,1}{0,0,0}{$\theta_3$}
        };
        \node[anchor=south west] at ({3*\sep}, {-1*\sep*\hs}) {
            \motzkinpath{skip,skip,skip}{0,1,1}{0,1,0}{$x_2\theta_3$}
        };
        \node[anchor=south west] at ({2*\sep}, {-2*\sep*\hs*\hs}) {
            \motzkinpath{skip,skip,skip}{0,0,1}{0,0,1}{$x_3\theta_2$}
        };
        \node[anchor=south west] at (3*\sep, {-2*\sep*\hs*\hs}) {
            \motzkinpath{skip,skip,skip}{0,0,0}{0,0,0}{$\theta_2\theta_3$}
        };
    \end{tikzpicture}
    \caption{The basis $B_3^{(1,1)}$. The $\alpha$-sequence is shown as the outline of all boxes, and those $x_i$ used for a particular basis element are shaded in gray.}
    \label{fig:super_Artin_3_example}
\end{figure}

We assume familiarity with the basics of symmetric function theory (see for example \cite[Chapter I]{Macdonald} or \cite[Chapter 7]{StanleyEC2}). Let $m_\lambda$, $e_\lambda$, $h_\lambda$, $p_\lambda$, and $s_\lambda$ denote respectively the monomial, elementary, complete homogeneous, power-sum, and Schur symmetric functions in infinitely many variables. 

Finally, we record the definitions of the multigraded Hilbert and Frobenius series of $R_n^{(2,2)}$ (see for example \cite{Bergeron2020}). For $R_n^{(k,j)}$ with $k,j \leq 2$, we can further specialize the following. Setting $t=0$ gives us the series for $(1,2)$, etc.
$R_n^{(2,2)}$ decomposes as a direct sum of multihomogenous components, which are $\mathfrak{S}_n$-modules:
\[R_n^{(2,2)} = \bigoplus_{r_1,r_2,s_1,s_2 \geq 0} (R_n^{(2,2)})_{r_1,r_2,s_1,s_2}.\]
We denote the multigraded Hilbert series by 
\[ \Hilb(R_n^{(2,2)}; q,t; u,v) := \sum_{r_1,r_2,s_1,s_2 \geq 0}\dim \left((R_n^{(2,2)})_{r_1,r_2,s_1,s_2} \right)q^{r_1}t^{r_2}u^{s_1}v^{s_2},\]
and the multigraded Frobenius series by 
\[ \Frob(R_n^{(2,2)}; q,t; u,v) := \sum_{r_1,r_2,s_1,s_2 \geq 0} F\Char\left((R_n^{(2,2)})_{r_1,r_2,s_1,s_2} \right)q^{r_1}t^{r_2}u^{s_1}v^{s_2},\]
where $F$ denotes the Frobenius characteristic map and $\Char$ denotes the character.
Recall that $\langle \Frob(R_n^{(2,2)}; q,t; u,v), h_1^n \rangle = \Hilb(R_n^{(2,2)}; q,t; u,v)$.

\section{The conjectural monomial basis}\label{sec:thebasis}

The goal of this section is to interpolate between the Kim--Rhoades basis and the super-Artin basis to construct a new set $B_n^{(1,2)}$, which we conjecture to be a basis for $R_n^{(1,2)}$.

We generalize the $\alpha$-sequence as follows. Let $\xi_S$ denote the ordered product $\xi_{s_1} \cdots \xi_{s_k}$ for any subset $S = \{ s_1 < \cdots < s_k\} \subseteq \{1,\ldots,n\}$. For any $T,S \subseteq \{2,\ldots,n\}$, define the \textit{generalized $\alpha$-sequence} $\alpha(T,S) = (\alpha_1(T,S), \ldots, \alpha_n(T,S))$ recursively by the initial condition $\alpha_1(T,S) = 0$ and for $2 \leq  i \leq n$, 
\[
    \alpha_i(T,S) =\alpha_{i-1}(T,S) -1 + \chi(i \not \in T) + \chi(i \not \in S).
\]

\begin{definition}\label{def:B_n^{(1,2)}}
     We let \[B_n^{(1,2)} := \{x^\alpha \theta_T \xi_S \, | \, \theta_T \xi_S \in B_n^{(0,2)} \text{ and } 0 \leq \alpha_i \leq \alpha_i(T,S) \text{ for all } i \in \{1,\ldots,n\}\}.\]
\end{definition}

See \Cref{fig:Basis_3_example} for an example of $B_n^{(1,2)}$ at $n=3$.

\begin{figure}[!htb]
    \centering
    \begin{tikzpicture}
        \def\sep{2.8}
        \node[anchor=south west] at (0, 0) {
            \motzkinpath{up,up,up}{0,1,2}{0,0,0}{1}
        };
        \node[anchor=south west] at ({\sep}, 0) {
            \motzkinpath{up,up,up}{0,1,2}{0,0,1}{$x_3$}
        };
        \node[anchor=south west] at ({2*\sep}, 0) {
            \motzkinpath{up,up,up}{0,1,2}{0,1,0}{$x_2$}
        };
        \node[anchor=south west] at ({3*\sep}, 0) {
            \motzkinpath{up,up,up}{0,1,2}{0,1,1}{$x_2x_3$}
        };
        \node[anchor=south west] at ({4*\sep}, 0) {
            \motzkinpath{up,up,up}{0,1,2}{0,0,2}{$x_3^2$}
        };
        \node[anchor=south west] at ({5*\sep}, 0) {
            \motzkinpath{up,up,up}{0,1,2}{0,1,2}{$x_2x_3^2$}
        };
        \node[anchor=south west] at (4*\sep, {-2*\sep}) {
            \motzkinpath{up,flat-theta,up}{0,0,1}{0,0,0}{$\theta_2$}
        };
        \node[anchor=south west] at ({\sep}, {-1*\sep}) {
            \motzkinpath{up,up,flat-theta}{0,1,1}{0,0,1}{$x_3\theta_3$}
        };
        \node[anchor=south west] at ({0*\sep}, {-3*\sep}) {
            \motzkinpath{up,flat-xi,up}{0,0,1}{0,0,0}{$\xi_2$}
        };
        \node[anchor=south west] at ({3*\sep}, {-1*\sep}) {
            \motzkinpath{up,up,flat-theta}{0,1,1}{0,1,1}{$x_2x_3\theta_3$}
        };
        \node[anchor=south west] at ({5*\sep}, {-1*\sep}) {
            \motzkinpath{up,up,flat-xi}{0,1,1}{0,0,1}{$x_3\xi_3$}
        };
        \node[anchor=south west] at ({1*\sep}, {-2*\sep}) {
            \motzkinpath{up,up,flat-xi}{0,1,1}{0,1,1}{$x_2x_3\xi_3$}
        };
        \node[anchor=south west] at (0, {-1*\sep}) {
            \motzkinpath{up,up,flat-theta}{0,1,1}{0,0,0}{$\theta_3$}
        };
        \node[anchor=south west] at ({4*\sep}, {-1*\sep}) {
            \motzkinpath{up,up,flat-xi}{0,1,1}{0,0,0}{$\xi_3$}
        };
        \node[anchor=south west] at ({2*\sep}, {-1*\sep}) {
            \motzkinpath{up,up,flat-theta}{0,1,1}{0,1,0}{$x_2\theta_3$}
        };
        \node[anchor=south west] at ({0*\sep}, {-2*\sep}) {
            \motzkinpath{up,up,flat-xi}{0,1,1}{0,1,0}{$x_2\xi_3$}
        };
        \node[anchor=south west] at ({5*\sep}, {-2*\sep}) {
            \motzkinpath{up,flat-theta,up}{0,0,1}{0,0,1}{$x_3\theta_2$}
        };
        \node[anchor=south west] at ({1*\sep}, {-3*\sep}) {
            \motzkinpath{up,flat-xi,up}{0,0,1}{0,0,1}{$x_3\xi_2$}
        };
        \node[anchor=south west] at (2*\sep, {-3*\sep}) {
            \motzkinpath{up,flat-theta,flat-theta}{0,0,0}{0,0,0}{$\theta_2\theta_3$}
        };
        \node[anchor=south west] at ({2*\sep}, {-2*\sep}) {
            \motzkinpath{up,up,down}{0,1,0}{0,0,0}{$\theta_3\xi_3$}
        };
        \node[anchor=south west] at ({4*\sep}, {-3*\sep}) {
            \motzkinpath{up,flat-xi,flat-theta}{0,0,0}{0,0,0}{$\xi_2\theta_3$}
        };
        \node[anchor=south west] at ({3*\sep}, {-2*\sep}) {
            \motzkinpath{up,up,down}{0,1,0}{0,1,0}{$x_2\theta_3\xi_3$}
        };
        \node[anchor=south west] at ({3*\sep}, {-3*\sep}) {
            \motzkinpath{up,flat-theta,flat-xi}{0,0,0}{0,0,0}{$\theta_2\xi_3$}
        };
        \node[anchor=south west] at ({5*\sep}, {-3*\sep}) {
            \motzkinpath{up,flat-xi,flat-xi}{0,0,0}{0,0,0}{$\xi_2\xi_3$}
        };
    \end{tikzpicture}
    \caption{The basis $B_3^{(1,2)}$. Below each modified Motzkin path is the outline of the generalized $\alpha$-sequence, and those $x_i$ used for a particular basis element are shaded in gray. Note that the outline of the generalized $\alpha$-sequence can be determined by the following rule: in each position $i$, the column of boxes extends up until its right side is one unit below the path above it.}
    \label{fig:Basis_3_example}
\end{figure}

We are ready to present our main conjecture.

\begin{conjecture}\label{conj:R_n^{(1,2)}}
    The set $B_n^{(1,2)}$ is a basis for $R_n^{(1,2)}$.
\end{conjecture}

\begin{remark}
$B_n^{(1,2)}$ specializes to the super-Artin basis $B_n^{(1,1)}$ by setting all $\xi_i = 0$, and $B_n^{(1,2)}$ specializes to the Kim--Rhoades basis $B_n^{(0,2)}$ by setting all $x_i = 0$.
\end{remark}

Next, we consider the implications of \Cref{conj:R_n^{(1,2)}} on the Hilbert series of $R_n^{(1,2)}$. Define 
\[\stair_q(\pi) := \prod_{k \in \alpha(T(\pi),S(\pi))} [k+1]_q,\]
where $T(\pi)$ and $S(\pi)$ are determined by which elements in $\{2,\ldots,n\}$ appear as indices for $\theta_i$ and $\xi_i$ respectively in the weight of the modified Motzkin path $\pi$.

\begin{example}
For the modified Motzkin path
$\pi = \motzkin{up,up,flat-theta}{}, $\\
we have that $T(\pi) = \{3\}$ and $S(\pi) = \varnothing$, so $\alpha(\{3\},\varnothing) = (0,1,1)$. Thus $\stair_q(\pi) = [1]_q[2]_q[2]_q$. Observe in \Cref{fig:Basis_3_example} that there are $4$ basis elements corresponding to $\pi$.
\end{example}

For a modified Motzkin path $\pi$, let $\deg_\theta(\pi) = |T(\pi)|$ and let $\deg_\xi(\pi) = |S(\pi)|$.
 
\begin{proposition}\label{prop:Hilb(R_n^{(1,2)})}
    Assuming \Cref{conj:R_n^{(1,2)}}, it follows that the Hilbert series of $R_n^{(1,2)}$ is 
    \[ \Hilb(R_n^{(1,2)};q;u,v) = \sum_{\pi \in \Pi(n)_{>0}} u^{\deg_\theta(\pi)} v^{\deg_\xi(\pi)} \stair_q(\pi).\]
\end{proposition}

We show that the proposed basis $B_n^{(1,2)}$ has Zabrocki's conjectured dimension, using a similar argument to Corteel--Nunge~\cite[Lemma 17]{CorteelNunge2020} which enumerates marked Laguerre histories.

\begin{theorem}\label{thm:cardinality}
    The cardinality of $B_n^{(1,2)}$ is $2^{n-1}n!$.
\end{theorem}

\section{A conjectural Frobenius series}\label{sec:conj_frob}

The goal of this section is to demonstrate how the conjectural basis can be used to propose a Frobenius series for $R_n^{(1,2)}$.

For any subset $S \subseteq \{1, \ldots, n-1\}$, the \textit{fundamental quasisymmetric function} $Q_{S,n}$ is defined by 
\[ Q_{S,n} = \sum_{\substack{a_1 \leq a_2 \leq \cdots \leq a_n,\\ a_i < a_{i+1} \text{ if } i \in S}} z_{a_1}z_{a_2}\cdots z_{a_n}.\]
We need the following definitions, which are motivated by related definitions of Iraci, Nadeau, and Vanden Wyngaerd on segmented permutations. 
For any $b \in B_n^{(1,2)}$, write
\[ b = \pm \prod_{i=1}^n x_i^{\alpha_i} \theta_i^{\beta_i} \xi_i^{\gamma_i},\]
for some exponents $\alpha_i \in \Z_{\geq 0}$ and $\beta_i,\gamma_i \in \{0,1\}$.
Since this is an ordered product, reordering the factors may change the sign, however, for our purposes the sign does not matter.
Then define $i$ to be an \textit{ascent of $b$} if and only if one of the following occurs:
    \begin{itemize}
        \item $\beta_i < \beta_{i+1}$;
        \item $\beta_i = \beta_{i+1} = 1$ and $\alpha_i \geq \alpha_{i+1} + \gamma_{i+1}$; or
        \item $\beta_i = \beta_{i+1} = 0$ and $\alpha_i < \alpha_{i+1} + \gamma_{i+1}$.
    \end{itemize}
    For $b \in B_{n}^{(1,2)}$, we say that 
    $\Asc(b) := \{ i \in \{1,\ldots,n-1\} \ | \ i \text{ is an ascent of } b\}.$
\begin{example}
    Consider $b = x_2x_4^2\theta_4\theta_5\xi_5 \in B_5^{(1,2)}$, which has $\alpha_2 = 1$, $\alpha_4=2$, $\beta_4 = 1$, $\beta_5 = 1$, $\gamma_5 = 1$, and all other exponents are $0$. This has ascents at $i=1$ (since $\beta_1 = \beta_2 = 0$ and $\alpha_1 < \alpha_2 + \gamma_2$), at $i=3$ (since $\beta_3 < \beta_4$), and at $i=4$ (since $\beta_4 = \beta_5 = 1$ and $\alpha_4 \geq \alpha_5 + \gamma_5$). Thus $\Asc(b) = \{1,3,4\}$.
\end{example}
Now we can state a conjectural Frobenius series for $R_n^{(1,2)}$ in terms of the set $B_n^{(1,2)}$.

\begin{conjecture}\label{conj:frob}
    \[\Frob(R_n^{(1,2)};q;u,v) = \sum_{b \in B_n^{(1,2)}} u^{\deg_\theta(b)} v^{\deg_\xi(b)} q^{\deg_x(b)} Q_{\Asc(b),n}.\]
\end{conjecture}

We will see further evidence for this conjecture with \Cref{thm:equiv_Frobenius}.


\section{The proposed basis and segmented permutations}\label{sec:everything-else}

In this section, we establish a $q,u,v$-weight preserving bijection between our proposed basis $B_n^{(1,2)}$ and the set of segmented permutations $\SW(1^n)$.
A \textit{segmented permutation} $\sigma$ is a permutation of $\{1,\ldots,n\}$, where between any letters, there may be a vertical bar $|$. 
For example, the segmented permutations of length $2$ are $12, 1|2, 21, 2|1$. 
Denote the set of all segmented permutations on $\{1,\ldots,n\}$ by $\SW(1^n)$.\footnote{The notation $\SW(1^n)$ comes from the more general class of \textit{segmented Smirnov words}, of which segmented permutations form a proper subset.} 
The cardinality of $\SW(1^n)$ is $2^{n-1}n!$.

Define a map $\psi: B_n^{(1,2)} \longrightarrow \SW(1^n)$ as follows. For any $b \in B_n^{(1,2)}$, we write it as
\[ b = \pm \prod_{i=1}^n x_i^{\alpha_i} \theta_i^{\beta_i} \xi_i^{\gamma_i},\]
for some $\alpha_i \in \Z_{\geq 0}$ and $\beta_i,\gamma_i \in \{0,1\}$. 
We always have $\alpha_1 = \beta_1 = \gamma_1 = 0$; start the corresponding segmented permutation with $1$. 
Then, as $i$ ranges from $2$ up through $n$, do exactly one of the following for each $i$:
\begin{enumerate}[label = (\alph*)]
    \item if $\beta_i = \gamma_i = 0$: insert ``$|i$'' or ``$i|$'' in such a way as to create a new block consisting of only $i$ at position $\alpha_i + 1$ from the rightmost block in the permutation (indexing starting at 1);
    \item if $\beta_i = 1$ and $\gamma_i = 0$: insert $i$ as the last element of an existing block, at position $\alpha_i + 1$ from the rightmost block in the permutation;
    \item if $\beta_i = 0$ and $\gamma_i = 1$: insert $i$ as the first element of an existing block, at position $\alpha_i + 1$ from the rightmost block in the permutation;
    \item if $\beta_i = \gamma_i = 1$: insert $i$ to replace a ``$|$'' and thus merge two adjacent blocks into one block, which is now at position $\alpha_i + 1$ from the rightmost block in the permutation.
\end{enumerate}
Upon completing this process, the output is some segmented permutation $\sigma$ in $\SW(1^n)$.

\begin{example}
    Consider $b = x_2x_4^2\theta_4\theta_5\xi_5$, which has $\alpha_2 = 1$, $\alpha_4=2$, $\beta_4 = 1$, $\beta_5 = 1$, $\gamma_5 = 1$, and all other exponents are $0$. Start building a segmented permutation with $1$. At $i=2$, we are in case (a), so we insert $2|$ to get $2|1$, so that $2$ is in a new block at position $1+1 = 2$ from the right. At $i=3$, we are in case (a), so we insert $|3$ to get $2|1|3$, so that $3$ is in a new block at the right. At $i=4$, we are in case (b), so we insert $4$ as the last element of the block at position $2+1=3$ from the right, giving $24|1|3$. At $i=5$, we are in case (d), so we insert $5$ to replace a ``$|$'' and merge two blocks which is now in the rightmost position, giving $24|153$.
\end{example}

In a segmented permutation $\sigma$, an \textit{ascent} (resp. \textit{descent}) is an index $i$ such that $\sigma_i < \sigma_{i+1}$ (resp. $\sigma_i > \sigma_{i+1}$) and there is no vertical bar $|$ between $\sigma_i$ and $\sigma_{i+1}$.
A consequence of the bijection is the following, where statistics $\sminv(\sigma)$ and $\Split(\sigma)$ on segmented permutations are defined in \cite{IraciNadeauVandenWyngaerd2024}.

\begin{theorem}\label{thm:bijection}
    The map $\psi : B_n^{(1,2)} \longrightarrow \SW(1^n)$ is a $q,u,v$-weight preserving bijection. For any $b \in B_n^{(1,2)}$, we have $\deg_\theta(b) = k$, the number of ascents in $\psi(b)$, $\deg_\xi(b) = \ell$, the number of descents in $\psi(b)$, $\deg_x(b) = \sminv(\psi(b))$, and $\Asc(b) = \Split(\psi(b))$.
\end{theorem}

Let $\SW(1^n,k,\ell)$ denote the set of segmented permutation of length $n$ with exactly $k$ ascents and $\ell$ descents.
Now, we are able to establish the equivalence of our conjectural Frobenius series (\Cref{conj:frob}) with a conjectural Frobenius series of Iraci, Nadeau, and Vanden Wyngaerd. This can also be interpreted as a symmetric function identity.

\begin{theorem}\label{thm:equiv_Frobenius}
    \[\sum_{b \in B_n^{(1,2)}} u^{\deg_\theta(b)} v^{\deg_\xi(b)} q^{\deg_x(b)} Q_{\Asc(b),n} = \sum_{ k+\ell < n} u^k v^\ell \sum_{\sigma \in \SW(1^n, k,\ell)} q^{\sminv(\sigma)} Q_{\Split(\sigma),n}.\]
\end{theorem}

Define the symmetric function $\SF(n,k,\ell) = \sum_{\sigma \in \SW(1^n, k,\ell)} q^{\sminv(\sigma)} Q_{\Split(\sigma),n}$ by the inner sum on the right hand side of \Cref{thm:equiv_Frobenius}
%, that is, $\sum_{\sigma \in \SW(1^n, k,\ell)} q^{\sminv(\sigma)} Q_{\Split(\sigma),n}$.
For a composition $\alpha \vDash n$, let $\Set({\alpha}) := (\alpha_1, \alpha_1 + \alpha_2, \ldots, \alpha_1 + \cdots + \alpha_{\ell-1})$. We have a formula for the $m_\mu$ coefficient of $\SF(n,k,\ell)$.

\begin{theorem}\label{thm:quasisymmetric-refinement-h_mu-basis} Let $\mu \vdash n$. For any fixed $k, \ell$, we have that
    \[ \langle \SF(n,k,\ell), h_\mu \rangle = \sum_{\substack{b \in B_n^{(1,2)},\\
    \deg_\theta(b) = k,\\
    \deg_\xi(b) = \ell,\\
    \Asc(b) \subseteq \Set(\mu)}} q^{\deg_x(b)}.\]
\end{theorem}

We derive the following formula for hook Schur functions $s_{(d+1,1^{n-d-1})}$, using the combinatorics of our conjectural basis. This generalizes a result on column Schur  functions $s_{(1^n)}$ \cite[Theorem 5.6]{IraciNadeauVandenWyngaerd2024}, which is recovered at $d=0$. To get the complete conjectural sign character, multiply by $u^kv^\ell$ and sum over all $k+\ell < n$.

\begin{theorem}\label{thm:qbinomial} 
For $0 \leq d \leq n-1$,
    \[\langle \SF(n,k,\ell) , s_{(d+1,1^{n-d-1})}\rangle = q^{\binom{n-d-k-\ell}{2}}\qbinom{n-1-d}{\ell}_q\qbinom{n-1-k}{d}_q\qbinom{n-1-\ell}{k}_q.\]
\end{theorem}


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\acknowledgements{The author would like to thank Fran\c{c}ois Bergeron, Sylvie Corteel, Nicolle Gonz\'alez, Sean Griffin, Mark Haiman, Brendon Rhoades, Josh Swanson, and Mike Zabrocki for invaluable conversations. }

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