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\articlenumber{87}
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\newtheorem{prop}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}
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\newtheorem{rmk}[thm]{Remark}
\newtheorem{conj}[thm]{Conjecture}

\newcommand{\CC}{{\mathbb {C}}}
\newcommand{\CCC}{{\mathcal {C}}}
\newcommand{\QQ}{{\mathbb {Q}}}
\newcommand{\PP}{{\mathbb {P}}}
\newcommand{\Frob}{{\mathrm {Frob}}}
\newcommand{\sign}{{\mathrm {sign}}}
\newcommand{\GL}{{\mathrm {GL}}}
\newcommand{\ZZ}{{\mathbb {Z}}}
\newcommand{\cont}{{\mathrm {cont}}}
\newcommand{\ch}{{\mathrm {ch}}}
\newcommand{\Hilb}{{\mathrm {Hilb}}}
\newcommand{\symm}{{\mathfrak {S}}}

\newcommand{\EEE}{{\mathcal {E}}}
\newcommand{\OP}{{\mathcal {OP}}}
\newcommand{\grFrob}{{\mathrm {grFrob}}}
\newcommand{\wt}{{\mathrm {wt}}}
\newcommand{\xx}{{\mathbf {x}}}
\newcommand{\coinv}{{\mathrm {coinv}}}


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%\usepackage{lipsum}

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\title{Harmonic bases for generalized coinvariant algebras}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Brendon Rhoades\thanks{\href{mailto:bprhoades@math.ucsd.edu}{bprhoades@math.ucsd.edu}. B. Rhoades was partially supported by NSF Grants DMS-1500838 and
DMS-1953781.}\addressmark{1},
Tianyi Yu\thanks{\href{mailto:tiy059@math.ucsd.edu}{tiy059@math.ucsd.edu}}\addressmark{1}, \and
Zehong Zhao\thanks{\href{mailto:zez045@math.ucsd.edu}{zez045@math.ucsd.edu}}\addressmark{1}}

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\address{\addressmark{1}Department of Mathematics, University of California, San Diego}

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\received{\today}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{Let $k \leq n$ be nonnegative integers and let $\lambda$ be a partition of $k$.
S. Griffin recently introduced a quotient $R_{n,\lambda}$ of the polynomial ring 
$\QQ[x_1, \dots, x_n]$ in $n$ variables which simultaneously generalizes the 
Delta Conjecture coinvariant rings of Haglund--Rhoades--Shimozono
and the cohomology rings of Springer fibers studied by Tanisaki and Garsia--Procesi.
We describe the space $V_{n,\lambda}$ of harmonics attached to $R_{n,\lambda}$ 
and produce a harmonic basis of $R_{n,\lambda}$ indexed by certain ordered set partitions $\OP_{n,\lambda}$.
Our description of $V_{n,\lambda}$ involves 
injective tableaux and Vandermonde determinants and
 combinatorics of our harmonic basis is governed by a new extension of 
the {\em Lehmer code} of a permutation to $\OP_{n, \lambda}$.}

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% \resume{\lipsum[2]}

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\keywords{coinvariant ring, harmonic space, Vandermonde}

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\begin{document}

\maketitle
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\section{Introduction: Quotient Rings and Harmonic Spaces}


In his Ph.D. thesis \cite{Griffin}, Sean Griffin introduced the following remarkable family of quotients of the polynomial
ring $\QQ[\xx_n] := \QQ[x_1, \dots, x_n]$ in $n$ variables.
Given a subset $S \subseteq [n] := \{1, 2, \dots, n \}$ and $d \geq 0$, let $e_d(S)$ be the 
degree $d$ elementary
symmetric polynomial in the variable set $\{ x_i \,:\, i \in S \}$.  For example, we have
\begin{equation*}
e_2(1457) = x_1 x_4 + x_1 x_5 + x_1 x_7 + x_4 x_5 + x_4 x_7 + x_5 x_7.
\end{equation*}
By convention, we set $e_d(S) = 0$ whenever $|S| < d$.

For $k \geq 0$,
in this paper we use the term {\em partition of $k$} to mean a weakly decreasing sequence
$\lambda = (\lambda_1 \geq \cdots \geq \lambda_s)$ of {\bf nonnegative} integers
with $\lambda_1 + \cdots + \lambda_s = k$.  We write $|\lambda| = k$ or $\lambda \vdash k$
to mean that $\lambda$ is a partition of $k$ and call $s$ the number of {\em parts} of 
$\lambda = (\lambda_1, \dots, \lambda_s)$.
%We also write $\ell(\lambda)$ for the number of {\em nonzero} parts of $\lambda$.
For example, if $\lambda = (4,2,2,0,0)$ we have $\lambda \vdash 8$ and the partition $\lambda$
has $5$ parts.


\begin{defn}[Griffin \cite{Griffin}]
  \label{quotient-definition}
  Let $k \leq n$ be nonnegative integers and let $\lambda = (\lambda_1 \geq \cdots \geq \lambda_{s} \geq 0)$
be a partition of $k$ with $s$ parts.
Write $\lambda' = (\lambda'_1 \geq \cdots \geq \lambda'_n \geq 0)$ for the conjugate partition of 
$\lambda$, padded with trailing zeros to have $n$ parts.

  Let $I_{n,\lambda} \subseteq \QQ[\xx_n]$ be the ideal
  \begin{equation}
  I_{n,\lambda} = \langle x_1^{s}, \dots, x_n^{s} \rangle + 
  \langle e_d(S) \,:\, S \subseteq [n] \text{ and } d > |S| - \lambda'_n - \lambda'_{n-1} - \cdots - \lambda'_{n-|S|+1} \rangle
  \end{equation}
  and let $R_{n,\lambda} := \QQ[\xx_n]/I_{n,\lambda}$ be the associated quotient ring
 % \footnote{Our notation differs from that of Griffin \cite{Griffin}. He considers partitions to only 
 % have positive parts and includes $s \geq \ell(\lambda)$ as a parameter, so that his objects 
 % are denoted $I_{n,\lambda,s}$ and $R_{n,\lambda,s}$. Griffin uses the symbol $I_{n,\lambda}$ to denote
 % the ideal $\langle e_d(S) \,:\, S \subseteq [n] \text{ and } d > |S| - \lambda'_n - \lambda'_{n-1} - \cdots - \lambda'_{n-|S|+1} \rangle$ and $R_{n,\lambda}$ to denote the corresponding quotient - this is the `limit' as $s \rightarrow \infty$ of
 % his $I_{n,\lambda,s}$ and $R_{n,\lambda,s}$.}.
  \end{defn}



  As an example, suppose $n = 9$ and $\lambda = (3,2,2,0)$ so that $k = 7$ and $s = 4$.
  The conjugate partition $\lambda' = (\lambda'_1, \dots, \lambda'_9)$ is given by
  $(3,3,1,0,0,0,0,0,0)$.
  The ideal $I_{9,\lambda} \subseteq \QQ[\xx_9]$ is generated by $x_1^4, \dots, x_9^4$ together with
  the polynomials
  \begin{align*}
  &e_i(S) \quad \quad S \subseteq [9] \quad \quad |S| = 9 \quad \quad i = 9,8,7,6,5,4,3,  \\
  &e_j(T) \quad \quad T \subseteq [9] \quad \quad |T| = 8 \quad \quad j = 8,7,6,5, \\
  &e_{d}(U) \quad \quad U \subseteq [9] \quad \quad |U| = 7 \quad \quad d = 7.
  \end{align*}
  Griffin's rings $R_{n,\lambda}$  generalize several important classes of quotient rings 
  in algebraic combinatorics.
  \begin{itemize}
  \item When $k = s = n$ and $\lambda = (1^n)$, the ideal $I_{n,\lambda}$ is generated by the 
  $n$ elementary symmetric polynomials $e_1(\xx_n), e_2(\xx_n), \dots, e_n(\xx_n)$ in the full variable set 
  $\{x_1, \dots, x_n \}$
  and $R_{n,\lambda}$ is the classical {\em coinvariant ring} 
$ R_n := \QQ[\xx_n]/ \langle e_1(\xx_n), e_2(\xx_n), \dots, e_n(\xx_n) \rangle$
  attached to the symmetric group $\symm_n$.
  The ring $R_{n,\lambda}$ presents the cohomology of the complete flag variety of type A$_{n-1}$.
  \item When $k = n$ and $\lambda \vdash n$ is arbitrary,
  the ring $R_{n,\lambda}$ is the {\em Tanisaki quotient} studied
  by Tanisaki \cite{Tanisaki} and Garsia--Procesi \cite{GP} which presents the cohomology of the
  {\em Springer fiber} $\mathcal{B}_{\lambda}$ attached to the partition $\lambda$. 
  \item When $\lambda = (1^k, 0^{s-k})$ has all parts less or equal to $1$, the rings $R_{n,\lambda}$ were 
  introduced by Haglund, Rhoades, and Shimozono \cite{HRS} to give a representation-theoretic model for
  the
  Haglund--Remmel--Wilson {\em Delta Conjecture} \cite{HRW}. 
 Pawlowski--Rhoades proved that these rings present the cohomology of
a certain moduli space $X_{n,k,s}$ of line configurations
 \cite{PR}.
  \end{itemize}
  
  
  The symmetric group $\symm_n$ acts on $\QQ[\xx_n]$ by subscript permutation.
  The ideals $I_{n,\lambda}$ are graded and $\symm_n$-stable, so $R_{n,\lambda}$ is a graded
  $\symm_n$-module.
  Generalizing results from \cite{GP, HRS}, Griffin calculated \cite{Griffin} the graded $\symm_n$-isomorphism
  type of $R_{n,\lambda}$.
 
  
  
In this extended abstract we study the rings $R_{n,\lambda}$ as graded $\QQ$-vector spaces.  In the special case
$k = s = n$ and $\lambda = (1^n)$, the classical coinvariant ring $R_n$
has a number of interesting bases which are important for different reasons.
Perhaps the simplest of these was discovered by E. Artin \cite{Artin}, who used Galois Theory to prove that 
the family of `sub-staircase monomials'
\begin{equation}
\{ x_1^{c_1} x_2^{c_2} \cdots x_n^{c_n} \,:\, 0 \leq c_i \leq n - i \}
\end{equation}
descends to a basis for $R_n$.
Extending earlier results of \cite{GP, HRS}, Griffin discovered the appropriate
generalization of `sub-staircase' to obtain a monomial basis
of $R_{n,\lambda}$; his result is quoted in Theorem~\ref{griffin-basis-theorem} below. 

Our main goal is to describe the {\em harmonic space} of the quotient ring $R_{n,\lambda}$
and so derive a {\em harmonic basis} of this quotient ring.
In order to motivate harmonic spaces and bases, we recall some  technical issues
that  arise in the study of  quotient rings.
  
  


Let  $I \subseteq \QQ[\xx_n]$
be any homogeneous ideal with  quotient ring $R = \QQ[\xx_n]/I$.
In algebraic combinatorics, 
one is often interested in calculating algebraic invariants of $R$ such as its  dimension or Hilbert
series.
A frequent impediment to computing these invariants is that, given $f \in \QQ[\xx_n]$, it can be difficult
to decide whether $f + I = 0$ in $R$.
Harmonic spaces can be used to replace quotients with subspaces, circumventing this problem.


For $f = f(x_1, \dots, x_n) \in \QQ[\xx_n]$, let $\partial f := f(\partial/\partial x_1, \dots, \partial/\partial x_n)$
be the differential operator on $\QQ[\xx_n]$ obtained by replacing each $x_i$ appearing in $f$ with the 
partial derivative $\partial/\partial x_i$.
The ring  $\QQ[\xx_n]$ acts on itself by
\begin{equation}
f \odot g := (\partial f)(g) \quad \text{for all $f, g \in \QQ[\xx_n]$.}
\end{equation}
That is, the polynomial $f \odot g$ is obtained by first turning $f$ into a differential operator $\partial f$,
and then applying $\partial f$ to $g$.

For $f, g \in \QQ[\xx_n]$, we define a number $\langle f, g \rangle \in \QQ$ by
\begin{equation}
\langle f, g \rangle := \text{constant term of $f \odot g$}.
\end{equation}
Then  $\langle -, - \rangle$ is an inner product on $\QQ[\xx_n]$ for which the set 
$\{ x_1^{a_1} \cdots x_n^{a_n} \,:\, a_i \geq 0 \}$ of monomials forms an orthogonal basis.

For a homogeneous ideal $I \subseteq \QQ[\xx_n]$, the {\em harmonic space} $V$ of $I$
is the graded subspace of $\QQ[\xx_n]$ given by the orthogonal complement
\begin{equation}
V = I^{\perp} = \{ g \in \QQ[\xx_n] \,:\, \langle f, g \rangle = 0 \text{ for all $f \in I$} \}.
\end{equation}
Writing $R = \QQ[\xx_n]/I$, standard results of linear algebra imply that
$\QQ[\xx_n] = V \oplus I$ so that any vector space basis for $V$ projects onto a basis of $R$.
Any basis of $V$ (and its image basis in $R$) is called a {\em harmonic basis}.
If the ideal $I$ is $\symm_n$-invariant, the $\symm_n$-invariance of the inner product
$\langle -, - \rangle$ furnishes an isomorphism of graded $\symm_n$-modules $R \cong V$.
Harmonic spaces are important for the following two reasons.
\begin{enumerate}
\item
As alluded to above, elements of the quotient ring $R$ are cosets whereas elements of the harmonic
space $V$ are honest polynomials.  Expensive calls to polynomial division inherent to
Buchberger-like algorithms involved in the study of $R$ are avoided when one uses the harmonic model $V$.
\item
It is often of interest to extend a quotient involving  $\xx$-variables $\{x_1, \dots, x_n\}$ 
to an additional set of $\mathbf{y}$-variables $\{y_1, \dots, y_n\}$. A key example is the 
extension of the classical coinvariant ring $R_n$ to the diagonal coinvariant ring $DR_n$.
While it can be unclear how to modify the ideal $I$ defining the above quotient ring $R$ to incorporate 
$\mathbf{y}$-variables in a natural way, the harmonic space $V$ can sometimes by so extended by closing
under the `polarization' operators $y_1 (\partial/\partial x_1) + \cdots + y_n (\partial/\partial x_n)$ together
with $\partial/\partial y_1, \dots, \partial/\partial y_n$.
\end{enumerate}


\begin{defn}
Let $V_{n,\lambda} \subseteq \QQ[\xx_n]$ be the harmonic space of $I_{n,\lambda}$.
\end{defn}


In the classical case $k = s = n$ and $\lambda = (1, \dots, 1)$ so that 
$R_{n,\lambda} = R_n$, the harmonic space has the following description.
Recall that the {\em Vandermonde determinant} $\delta_n \in \QQ[\xx_n]$ is the polynomial
\begin{equation}
\delta_n := \prod_{1 \leq i < j \leq n} (x_i - x_j).
\end{equation}
The harmonic space $V_n \subseteq \QQ[\xx_n]$ corresponding
to $R_n$ is generated by $\delta_n$ as a $\QQ[\xx_n]$-module.
Equivalently, the space $V_n$ is the smallest subspace of $\QQ[\xx_n]$ containing $\delta_n$
which is closed under the partial derivatives $\partial/\partial x_1, \dots, \partial/\partial x_n$.
A harmonic basis of $R_n$ is given by applying sub-staircase monomials (as differential operators)
to $\delta_n$:
\begin{equation}
\{ (x_1^{c_1} \cdots x_n^{c_n}) \odot \delta_n \,:\, 0 \leq c_i \leq n - i \}.
\end{equation}
In the Springer fiber case $k = n$ with $\lambda$ arbitrary,
the harmonic space $V_{n,\lambda}$ was described by N. Bergeron and Garsia \cite{BG} 
using `partial Vandermonde' polynomials.

The main results of this extended abstract are as follows.

\begin{itemize}
\item
We show that $V_{n,\lambda}$ is generated as a $\QQ[\xx_n]$-module by an explicit 
family $\delta_T$ of polynomials indexed by {\em injective tableaux}
$T$ of shape $\lambda$ with entries $\leq n$
(Theorem~\ref{generating-harmonic-set}).
\item
We describe a harmonic basis for $V_{n,\lambda}$ indexed by certain
{\em ordered set partitions}
(Theorem~\ref{harmonic-basis}).
\item
In order to describe the harmonic basis of Theorem~\ref{harmonic-basis}, we introduce a 
{\em coinversion statistic} $\coinv$ and an associated {\em coinversion code} $\code$
on ordered set partitions which generalize the classical Lehmer code on permutations
(Section~\ref{Coinversion}). The definition of our coinversion code is reminiscent of the 
diagonal inversion statistic
$\mathrm{dinv}$  in the theory of Macdonald polynomials \cite{HHL}.
The coinversion statistic may be used to calculate the Hilbert series of 
$R_{n,\lambda}$ (Corollary~\ref{hilbert-series}).
\end{itemize}







\section{Coinversion codes for ordered set partitions}
\label{Coinversion}


\subsection{Coinversion codes}
Given a partition $\lambda = (\lambda_1 \geq \cdots \geq \lambda_s) \vdash k$
and $n \geq k$, we let $\OP_{n,\lambda}$ be the family of sequences
$\sigma = (B_1, \dots, B_s)$ of subsets of $[n] := \{1, \dots, n \}$ such that
$[n] = B_1 \sqcup \cdots \sqcup B_s$ (disjoint union) and $|B_i| \geq \lambda_i$ for all $i$.
We call elements of $\OP_{n,\lambda}$ {\em ordered set partitions}, despite the fact that
when $\lambda_i = 0$ we allow the possibility that $B_i = \varnothing$.
For example, if $n = 16$ and $\lambda = (3,3,2,2,0,0)$ (so that $s = 6$) we have
\begin{equation}
\label{basic-example}
(\{1,3,5,9\}, \{6,7,8,10,14\}, \{2,12,15\}, \{4,13\}, \varnothing, \{11,16\}) \in \OP_{n,\lambda}.
\end{equation}


When $\lambda = (1^n) \vdash n$, we have a natural identification of ordered set partitions 
in $\OP_{n,\lambda}$ with permutations in $\symm_n$. 
The {\em coinversion number} $\coinv(w)$ of a permutation $w = w(1) \cdots w(n)$ in $\symm_n$
counts the number of noninversions in $w$, i.e.
\begin{equation}
\coinv(w) := | \{1 \leq i < j \leq n \,:\, w(i) < w(j) \} |.
\end{equation}
The {\em coinversion code} of $w$ is the sequence $(c_1, \dots, c_n)$ where
\begin{equation}
c_j := | \{ 1 \leq i < j \,:\, w(i) < w(j) \} |.
\end{equation}
This is a variant on the {\em Lehmer code} of the permutation $w$. The coinversion
code $(c_1, \dots, c_n)$ refines the coinversion number in the sense that 
$c_1 + \cdots + c_n = \coinv(w)$.  The map $w \mapsto (c_1, \dots, c_n)$
sending $w$ to its coinversion code gives a bijection from $\symm_n$ to the set
$[n-1] \times [n-2] \times \cdots \times [0]$ of sequences which fit below a staircase.
These substaircase sequences also index the Artin basis
$\{ x_1^{c_1} \cdots x_1^{c_n} \,:\, 0 \leq c_i \leq n - i \}$ of the classical coinvariant ring $R_n$.

Our first results extend the notion of coinversion number and coinversion code from 
$\symm_n$ to $\OP_{n,\lambda}$.
To define these statistics, we adopt a tableau-like representation of ordered set partitions
in $\OP_{n,\lambda}$.  
If $\lambda = (\lambda_1 \geq \cdots \geq \lambda_s)$ is a partition and 
$\sigma = (B_1, \dots, B_s) \in \OP_{n,\lambda}$, the {\em container diagram}
of $\sigma$ is obtained by 
\begin{itemize}
\item first placing from left to right $\lambda_i$ top-justified boxes in column $i$
(these boxes are called the {\em container}), and then
\item filling column $i$ with the elements of $B_i$ in increasing order from bottom to top,
where we will have entries which `float' above the boxes whenever $|B_i| > \lambda_i$.
\end{itemize}
For example, when $n = 16$ and $\lambda = (3,3,2,2,0,0)$, the container diagram of the ordered set partition 
$\sigma$ of 
\eqref{basic-example} is shown below.
\begin{equation}
\label{example-sigma}
\begin{Young}
 ,& ,14 & ,&,  &,  &,16  \cr
 ,9 & ,10 & ,15 &, &, \varnothing  &, 11 \cr
 5 & 8 & 12 & 13 &, &,   \cr
 3 & 7 & 2 & 4 &, &, \cr
1   & 6 &, &, &, &,  \cr 
\end{Young}
\end{equation}
Empty blocks in ordered set partitions give rise to empty columns in container diagrams.
The condition $|B_i| \geq \lambda_i$ corresponds to every box in the container being filled 
with a number.
Numbers which do not lie in the container are called {\em floating}.

Let $\lambda = (\lambda_1 \geq \cdots \geq \lambda_s)$ be a partition and let $\sigma \in \OP_{n,\lambda}$
be an ordered set partition, thought of in terms of its container diagram.  For $1 \leq i < j \leq n$, 
we say that the pair $(i,j)$ is a {\em coinversion} of $\sigma$ when one of the following three conditions
hold:
\begin{itemize}
\item $i$ is not floating, $j$ is to the right of $i$ in $\sigma$, $i$ and $j$ are in the same row of $\sigma$, and
$i < j$,
\item $i$ is not floating, $j$ is to the left of $i$ in $\sigma$, $j$ is one row below $i$ in $\sigma$, and
$i < j$, or
\item $i$ is floating, $j$ is to the right of $i$ in $\sigma$, $j$ is at the top of its container, and $i < j$.
\end{itemize}

The first two conditions may be depicted schematically as 
\begin{equation*}
\begin{Young}
i &, &, \cdots &, &  j 
\end{Young}  \hspace{0.2in}   \text{and}  \hspace{0.2in}
\begin{Young}
 &  ,&, \cdots &, &i \cr
 j
\end{Young}
\end{equation*}
where $i$ and $j$ are in adjacent rows in the diagram on the right.
The last condition may be depicted 
\begin{equation*}
\begin{Young}
,i &, &,  &, &, \cr
, & ,& , \ddots &, &,  \cr
 & , & ,  &, &j
 \end{Young}
\end{equation*}
where $j$ is at the top of the container and there is an arbitrary positive number of rows separating
$i$ and $j$.


\begin{rmk}
The conditions defining coinversions 
for non-floating indices are the same as those used to define the statistic $\mathrm{dinv}$ which arises
in the Haglund--Haiman--Loehr monomial expansion of the modified Macdonald polynomials \cite{HHL}.
\end{rmk}


For $1 \leq i \leq n$ we define a number $c_i \geq 0$ by
\begin{equation}
c_i := \begin{cases}
| \{ i < j \,:\, \text{$(i,j)$ is a coinversion of $\sigma$} \} |
& \text{if $i$ is not floating} \\
| \{ i < j \,:\, \text{$(i,j)$ is a coinversion of $\sigma$} \} | + (p-1) 
& \text{if $i$ is floating in the $p$\textsuperscript{th} block of $\sigma$.}
\end{cases}
\end{equation}
The {\em coinversion code} of $\sigma$ is given by
\begin{equation}
\code(\sigma) := (c_1, \dots, c_n)
\end{equation}
and the {\em coinversion number} of $\sigma$ is given by
\begin{equation}
\coinv(\sigma) := c_1 + \cdots + c_n.
\end{equation}
Rhoades and Wilson \cite{RW2} defined $\code(\sigma)$ in the special case
where $\lambda = (1^k)$.


As an example of these concepts, consider the ordered set partition
$\sigma \in \OP_{16,(3,3,2,2,0,0)}$ appearing in \eqref{example-sigma}.
Let $(c_1, \dots, c_{16})$ be the sequence $\code(\sigma)$.
The entry $2$ forms coinversions with $4$ and $6$, so that
$c_2 = 2$.
The entry 10 is floating in column 2, and forms coinversions with 12 and 13
so that $c_{10} = 2 + (2-1) = 3$.  We have
\begin{equation*}
\code(\sigma) = (c_1, \dots, c_{16}) = (1, 2, 2, 1, 3, 0, 0, 2, 2, 3, 5, 1, 0, 1, 2, 5).
\end{equation*}
Adding this sequence yields $\coinv(\sigma) = 30$.



\subsection{Shuffles and the code map}
The code map $\sigma \mapsto \code(\sigma)$ sends ordered set partitions in $\OP_{n,\lambda}$
to length $n$ sequences of nonnegative integers.
It turns out that this map is injective; to describe its image
we  need some definitions.

Recall that a {\em shuffle} of two sequences $(a_1, \dots, a_r)$ and $(b_1, \dots, b_s)$
is an interleaving $(c_1, \dots, c_{r+s})$ of these sequences which preserves the relative order
of the $a$'s and $b$'s. Analogously (or by induction), one can define a shuffle of any number of sequences.

Let $k \leq n$ be nonnegative integers and let
$\lambda = (\lambda_1 \geq \cdots \geq \lambda_s)$ be a partition of $k$ with $s$ nonnegative parts.
Write its conjugate as $\lambda' = (\lambda'_1 \geq \cdots \geq \lambda'_k)$. We define
$\CCC_{n,\lambda}$ to be the family of length $n$ sequences $(c_1, \dots, c_n)$ of nonnegative integers
which are componentwise $\leq$ some shuffle of the $k+1$ (possibly empty) sequences
\begin{equation*}
(\lambda'_1 - 1, \lambda'_1 - 2, \dots, 1, 0), \dots, (\lambda'_k - 1, \lambda'_k - 2, \dots, 1, 0), \text{ and }
(s-1, s-1, \dots, s-1)
\end{equation*}
where the final sequence has $n-k$ copies of $s-1$.

\begin{thm}
\label{code-is-bijection}
Let $k \leq n$ be positive integers and let $\lambda = (\lambda_1 \geq \cdots \geq \lambda_s) \vdash k$
be a partition of $k$. The map 
$\code: \OP_{n,\lambda} \rightarrow \CCC_{n,\lambda}$ is a bijection.
\end{thm}

\begin{proof} (Sketch) We describe the inverse map $\iota: \CCC_{n,\lambda} \rightarrow \OP_{n,\lambda}$
in terms of an insertion algorithm.  For any ordered set partition 
$(B_1, \dots, B_s)$, we place the blocks $B_1, \dots, B_s$ in the container
 diagram corresponding to $\lambda$, from left to right. For example, if   $\lambda = (3,3,2,2,0,0)$
 and
 \begin{equation*}
(B_1 , \cdots , B_s) = ( \{4\} , \{ 2,  3,  6 \} , \{1 \} , \varnothing,  \varnothing , \{ 5 \})
\end{equation*}
we obtain the diagram
\begin{equation*}
\begin{Young}
  , & , & , & , &, \varnothing & , 5 \cr
  & 6  &  &  &, &,   \cr
  & 3 & 1 &  &, &, \cr
 4  & 2  &, &, &, &,  \cr 
, \cr
,1 & ,3 & ,2 & ,0 & ,4 & ,5 
\end{Young}
\end{equation*}
where the first, third, and fourth columns from the left remain unfilled.
We label the columns of the diagram of $(B_1, \dots, B_s)$ with the $s$ 
distinct {\em coinversion labels} as follows. We first label the $a$ columns which
contain no empty boxes (including those columns corresponding to empty parts of $\lambda$)
from right to left with $s-1, s-2, \dots, s-a$, then label the $b$ columns which contain a single empty box
from left to right with $s-a-1, s-a-2, \dots, s-a-b$, then label the $c$ columns which contain two empty boxes
from left to right with $s-a-b-1, s-a-b-2, \dots, s-a-b-c$, and so on. Note that coinversion labels are 
assigned `in the opposite direction' when labeled filled and unfilled columns. The coinversion labels are displayed 
below the columns of the container diagram.

Suppose we have a sequence $(c_1, \dots, c_n) \in \OP_{n,\lambda}$. We define
$\iota(c_1, \dots, c_n)$ by starting with the `empty' diagram corresponding to $(\varnothing, \dots, \varnothing)$
and, for each $i = 1, 2, \dots, n$, inserting $i$ into the unique column with coinversion label $c_i$
(updating coinversion labels as we go). For example if $\lambda = (3,3,2,2,0,0)$ (so that $s = 6$) and
$n = 16$, the sequence 
\begin{equation*}
(c_1, \dots, c_{16}) = (1,2,2,1,3,0,0,2,2,3,5,1,0,1,2,5) \in \CCC_{16,\lambda}
\end{equation*}
inserts to
\begin{scriptsize}
\begin{equation*}
\begin{Young}
  , & , & , & , &, \varnothing & , \varnothing \cr
  &   &  &  &, &,   \cr
  &  &  &  &, &, \cr
   &   &, &, &, &,  \cr 
, \cr
,1 & ,0 & ,3 & ,2 & ,4 & ,5 
\end{Young}  \quad \quad 
\begin{Young}
  , & , & , & , &, \varnothing & , \varnothing \cr
  &   &  &  &, &,   \cr
  &  &  &  &, &, \cr
1   &   &, &, &, &,  \cr 
, \cr
,3 & ,0 & ,2 & ,1 & ,4 & ,5 
\end{Young}  \quad \quad
\begin{Young}
  , & , & , & , &, \varnothing & , \varnothing \cr
  &   &  &  &, &,   \cr
  &  & 2 &  &, &, \cr
1   &   &, &, &, &,  \cr 
, \cr
,2 & ,0 & ,3 & ,1 & ,4 & ,5 
\end{Young}  \quad \quad
\begin{Young}
  , & , & , & , &, \varnothing & , \varnothing \cr
  &   &  &  &, &,   \cr
3  &  & 2 &  &, &, \cr
1   &   &, &, &, &,  \cr 
, \cr
,3 & ,0 & ,2 & ,1 & ,4 & ,5 
\end{Young}  \quad \quad
\begin{Young}
  , & , & , & , &, \varnothing & , \varnothing \cr
  &   &  &  &, &,   \cr
3  &  & 2 & 4  &, &, \cr
1   &   &, &, &, &,  \cr 
, \cr
,3 & ,0 & ,2 & ,1 & ,4 & ,5 
\end{Young}  \quad \quad
\end{equation*}

\begin{equation*}
\begin{Young}
  , & , & , & , &, \varnothing & , \varnothing \cr
 5 &   &  &  &, &,   \cr
3  &  & 2 & 4  &, &, \cr
1   &   &, &, &, &,  \cr 
, \cr
,3 & ,0 & ,2 & ,1 & ,4 & ,5 
\end{Young}  \quad \quad
\begin{Young}
  , & , & , & , &, \varnothing & , \varnothing \cr
 5 &   &  &  &, &,   \cr
3  &  & 2 & 4  &, &, \cr
1   &  6 &, &, &, &,  \cr 
, \cr
,3 & ,0 & ,2 & ,1 & ,4 & ,5 
\end{Young}  \quad \quad
\begin{Young}
  , & , & , & , &, \varnothing & , \varnothing \cr
 5 &   &  &  &, &,   \cr
3  & 7 & 2 & 4  &, &, \cr
1   &  6 &, &, &, &,  \cr 
, \cr
,3 & ,2 & ,1 & ,0 & ,4 & ,5 
\end{Young}  \quad \quad
\begin{Young}
  , & , & , & , &, \varnothing & , \varnothing \cr
 5 & 8   &  &  &, &,   \cr
3  & 7 & 2 & 4  &, &, \cr
1   &  6 &, &, &, &,  \cr 
, \cr
,2 & ,3 & ,1 & ,0 & ,4 & ,5 
\end{Young}  \quad \quad
\begin{Young}
   , 9 & , & , & , &, \varnothing & , \varnothing \cr
 5 & 8   &  &  &, &,   \cr
3  & 7 & 2 & 4  &, &, \cr
1   &  6 &, &, &, &,  \cr 
, \cr
,2 & ,3 & ,1 & ,0 & ,4 & ,5 
\end{Young}  \quad \quad
\begin{Young}
   , 9 & , 10 & , & , &, \varnothing & , \varnothing \cr
 5 & 8   &  &  &, &,   \cr
3  & 7 & 2 & 4  &, &, \cr
1   &  6 &, &, &, &,  \cr 
, \cr
,2 & ,3 & ,1 & ,0 & ,4 & ,5 
\end{Young}  \quad \quad
\end{equation*}

\begin{equation*}
\begin{Young}
   , 9 & , 10 & , & , &, \varnothing & , 11 \cr
 5 & 8   &  &  &, &,   \cr
3  & 7 & 2 & 4  &, &, \cr
1   &  6 &, &, &, &,  \cr 
, \cr
,2 & ,3 & ,1 & ,0 & ,4 & ,5 
\end{Young}  \quad \quad
\begin{Young}
   , 9 & , 10 & , & , &, \varnothing & , 11 \cr
 5 & 8   &  12 &  &, &,   \cr
3  & 7 & 2 & 4  &, &, \cr
1   &  6 &, &, &, &,  \cr 
, \cr
,1 & ,2 & ,3 & ,0 & ,4 & ,5 
\end{Young}  \quad \quad
\begin{Young}
   , 9 & , 10 & , & , &, \varnothing & , 11 \cr
 5 & 8   &  12 & 13  &, &,   \cr
3  & 7 & 2 & 4  &, &, \cr
1   &  6 &, &, &, &,  \cr 
, \cr
,0 & ,1 & ,2 & ,3 & ,4 & ,5 
\end{Young}  \quad \quad
\begin{Young}
   , & , 14 &, &, &, &, \cr
   , 9 & , 10 & , & , &, \varnothing & , 11 \cr
 5 & 8   &  12 & 13  &, &,   \cr
3  & 7 & 2 & 4  &, &, \cr
1   &  6 &, &, &, &,  \cr 
, \cr
,0 & ,1 & ,2 & ,3 & ,4 & ,5 
\end{Young}  \quad \quad
\begin{Young}
   , & , 14 &, &, &, &, \cr
   , 9 & , 10 & , 15 & , &, \varnothing & , 11 \cr
 5 & 8   &  12 & 13  &, &,   \cr
3  & 7 & 2 & 4  &, &, \cr
1   &  6 &, &, &, &,  \cr 
, \cr
,0 & ,1 & ,2 & ,3 & ,4 & ,5 
\end{Young}  \quad \quad
\begin{Young}
   , & , 14 &, &, &, &, 16 \cr
   , 9 & , 10 & , 15 & , &, \varnothing & , 11 \cr
 5 & 8   &  12 & 13  &, &,   \cr
3  & 7 & 2 & 4  &, &, \cr
1   &  6 &, &, &, &,  \cr 
, \cr
,0 & ,1 & ,2 & ,3 & ,4 & ,5 
\end{Young}  \quad \quad
\end{equation*}
\end{scriptsize}
One verifies that $\iota$ is well-defined and that $\code$ and $\iota$ are mutually inverse.
\end{proof}

Theorem~\ref{code-is-bijection} is proven by constructing an insertion-style 
inverse map $\CCC_{n,\lambda} \rightarrow \OP_{n,\lambda}$.
The family of sequences $\CCC_{n,\lambda}$ arises algebraically in the study of 
$R_{n,\lambda}$. Griffin proved that $R_{n,\lambda}$ has the following Artin-like monomial
basis \cite{Griffin}.


\begin{thm}[Griffin \cite{Griffin}]
\label{griffin-basis-theorem}
Let $k \leq n$ be positive integers and let $\lambda = (\lambda_1 \geq \cdots \geq \lambda_s) \vdash k$.
The set of monomials
\begin{equation}
\{ x_1^{c_1} \cdots x_n^{c_n} \,:\, (c_1, \dots, c_n) \in \CCC_{n,\lambda} \}
\end{equation}
descends to a vector space basis of $R_{n,\lambda}$.
\end{thm}

Theorem~\ref{code-is-bijection} gives a direct combinatorial link between the monomial 
basis of Theorem~\ref{griffin-basis-theorem} and ordered set partitions.
Theorem~\ref{code-is-bijection} will also be useful in study of harmonic bases.








\section{The harmonic space $V_{n,\lambda}$}

\subsection{Injective tableaux and harmonic generators}

We regard $\QQ[\xx_n]$ as a module over itself by the differential operator action
$f \odot g := \partial f (g)$.
The harmonic space $V_{n,\lambda}$ of $R_{n,\lambda}$ forms a $\QQ[\xx_n]$-submodule of 
$\QQ[\xx_n]$. The submodule $V_{n,\lambda}$ is not principal, but we may describe a 
generating set combinatorially.

Let $\lambda$ be a partition. An {\em  tableau} of shape $\lambda$ is a filling 
$T: \lambda \rightarrow \{1, 2, \dots \}$ of the boxes of $\lambda$ with positive integers.
The tableau $T$ is {\em injective} if its entries are distinct and {\em column strict} if its entries
increase strictly going down columns. We let $\Inj(\lambda; \leq n)$ be the family of 
injective and column strict tableaux of shape $\lambda$ with entries $\leq n$.
A sample element of $\Inj((4,2,1,0,0); \leq 9)$ appears below; observe that the entries $7$ and $8$
do not appear in this tableau.
\begin{equation}
\label{example-injective-tableau}
\begin{Young}
2 & 1 & 3 & 9 \cr
5 & 4 \cr
6
\end{Young}
\end{equation}

Let $\lambda = (\lambda_1 \geq \cdots \geq \lambda_s)$ be a partition with $s$ parts.
We attach a polynomial $\delta_T \in \QQ[\xx_n]$ to any tableau $T \in \Inj(\lambda; \leq n)$ as follows.
For any subset $S \subseteq [n]$, we let $\delta_S := \prod_{\substack{i,j \in S \\ i < j}} (x_ i - x_j)$
be the Vandermonde determinant in the family of variables indexed by $S$. 
If the tableau $T$ has columns $C_1, \dots, C_r$ we set 
\begin{equation}
\delta_T := \delta_{C_1} \cdots \delta_{C_r} \times \prod_{i \notin T} x_i^{s-1}
\end{equation}
where the final product is over all indices $1 \leq i \leq n$ which do not appear in the tableau $T$.
For example, if $\lambda = (4,2,1,0,0)$ (so that $s = 5$) and $T$ is as in 
\eqref{example-injective-tableau} then
\begin{align*}
\delta_T &= \delta_{\{2,5,6\}} \times \delta_{\{1,4\}} \times \delta_{\{3\}} \times \delta_{\{9\}} \times
x_7^4 x_8^4 \\
&= (x_2 - x_5)(x_2 - x_6)(x_5 - x_6) \times (x_1 - x_4) \times 1 \times 1 \times x_7^4 x_8^4.
\end{align*}
The polynomials $\delta_T$ indexed by injective tableaux generate the harmonic space
$V_{n,\lambda}$.

\begin{thm}
\label{generating-harmonic-set}
Let $k \leq n$ be positive integers and let $\lambda = (\lambda_1 \geq \cdots \geq \lambda_s)$ be a partition
of $k$.  The harmonic space $V_{n,\lambda}$ is the smallest subspace of $\QQ[\xx_n]$ which
\begin{itemize}
\item contains the polynomial $\delta_T$ for every tableau $T \in \Inj(\lambda, \leq n)$, and
\item is closed under the partial derivative operators
$\partial/\partial x_1, \dots, \partial/\partial x_n$.
\end{itemize}
Equivalently, the set $\{ \delta_T \,:\, T \in \Inj(\lambda, \leq n) \}$ generates $V_{n,\lambda}$ 
as a $\QQ[\xx_n]$-module.
\end{thm}


Theorem~\ref{generating-harmonic-set} gives an effective way to realize the 
$\symm_n$-module $R_{n,\lambda}$ on a computer.
The use of harmonic polynomials in $V_{n,\lambda}$ avoids the computationally expensive
calls to the multivariate polynomial long division involved in Gr\"obner-theoretic algorithms
operating on the quotient $R_{n,\lambda} = \QQ[\xx_n]/I_{n,\lambda}$.
The description of $V_{n,\lambda}$ in terms of injective tableaux is also
arguably 
simpler than the quotient ring formulation of 
$R_{n,\lambda}$ in Definition~\ref{quotient-definition}.







\subsection{Harmonic bases}

Theorem~\ref{generating-harmonic-set} furnishes a natural algorithm spanning set of the quotient ring 
$R_{n,\lambda}$: starting with the family of polynomials 
$\{ \delta_T \,:\, T \in \Inj(\lambda, \leq n) \}$, apply all possible partial derivatives to obtain a set of polynomials
\begin{equation}
\label{spanning-set}
\{ (\partial/\partial x_1)^{b_1} \cdots (\partial/\partial x_n)^{b_n} \delta_T \,:\,
T \in \Inj(\lambda; \leq n), \, \, b_i \geq 0 \}
\end{equation}
which spans $R_{n,\lambda}$.
Since there are finitely many injective tableaux, the set \eqref{spanning-set} is finite.

It is natural to ask for a basis of $R_{n,\lambda}$ extracted from the spanning set \eqref{spanning-set}.
Such a basis may be obtained using container diagrams and coinversion codes.
Given a partition $\lambda = (\lambda_1, \dots, \lambda_s)$ of $n$ with $s$ parts,
we think of an ordered set partition $\sigma \in \OP_{n,\lambda}$
in terms of its container diagram. We define an injective tableau $T(\sigma) \in \Inj(\lambda; \leq n)$ by letting 
the $i^{th}$ column of $T$ be the entries in the $i^{th}$ row of the container of $\sigma$.
We also define a sequence $\maxcode(\sigma) = (a_1, \dots, a_n)$ of nonnegative integers by letting
\begin{equation}
a_i := \begin{cases}
b & \text{if $i$ appears in $T(\sigma)$ with $b$ boxes directly below it,} \\
s-1 & \text{if $i$ does not appear in $T(\sigma)$.}
\end{cases}
\end{equation}
We define the polynomial $\delta_{\sigma} \in \OP_{n,\lambda}$ by the rule
\begin{equation}
\delta_{\sigma} := (x_1^{a_1 - c_1} \cdots x_n^{a_n - c_n}) \odot \delta_{T(\sigma)}
\end{equation}
where $\code(\sigma) = (c_1, \dots, c_n)$ is the coinversion code of $\sigma$.

As an example of these concepts, if $n = 16$ and $\lambda = (3,3,2,2,0,0)$ (so that $s = 6)$ and
$\sigma$ is the ordered set partition of \eqref{example-sigma}
(rewritten below)
\begin{equation*}
\begin{Young}
,& ,& ,& ,14 & ,&,  &,  &,16  \cr
,& ,& ,9 & ,10 & ,15 &, &, \varnothing  &, 11 \cr
,\sigma = & ,& 5 & 8 & 12 & 13 &, &,   \cr
,& ,& 3 & 7 & 2 & 4 &, &, \cr
,& ,& 1   & 6 &, &, &, &,  \cr 
\end{Young}
\end{equation*}
which gives the injective tableau $T(\sigma) \in \Inj(\lambda; \leq n)$
\begin{equation*}
\begin{Young}
,& ,& ,& 5 & 2 & 1 \cr
,& ,& ,& 8 & 3 & 6 \cr
,T(\sigma) = & ,& ,&12 & 4 \cr
,& ,& ,& 13 & 7 \cr
\end{Young}
\end{equation*}
and the corresponding $\maxcode$ sequence 
\begin{equation*}
\maxcode(\sigma) = (a_1, \dots, a_{16}) = (1,3,2,1,3,0,0,2,5,5,5,1,0,5,5,5).
\end{equation*}
The coinversion code of $\sigma$ is 
\begin{equation*}
\code(\sigma) = (c_1, \dots, c_{16}) = (1,2,2,1,3,0,0,2,2,3,5,1,0,1,2,5)
\end{equation*}
and we have 
$\maxcode(\sigma) - \code(\sigma) = (0,1,0,0,0,0,0,0,3,2,0,0,0,4,3,0)$ so that 
\begin{equation*}
\delta_{\sigma} = (x_1^0 x_2^1 x_3^0 x_4^0 x_5^0 x_6^0 x_7^0 x_8^0 x_9^3 x_{10}^2
x_{11}^0 x_{12}^0 x_{13}^0 x_{14}^4 x_{15}^3 x_{16}^0) \odot \delta_{T(\sigma)}.
\end{equation*}

\begin{thm}
\label{harmonic-basis}
Let $k \leq n$ be positive integers and let $\lambda = (\lambda_1 \geq \cdots \geq \lambda_s)$ be a partition
of $k$.  The set 
\begin{equation}
\{ \delta_{\sigma} \,:\, \sigma \in \OP_{n,\lambda} \}
\end{equation}
is a harmonic basis of $R_{n,\lambda}$. The lexicographical leading term of $\delta_{\sigma}$
has exponent sequence $\code(\sigma)$.
\end{thm}


It follows from the definitions that the degree of the polynomial $\delta_{\sigma}$
is the coinversion number $\coinv(\sigma)$. Theorem~\ref{harmonic-basis}
therefore gives a combinatorial formula for the Hilbert series of $R_{n,\lambda}$.


\begin{cor}
\label{hilbert-series}
Let $k \leq n$ be positive integers and let $\lambda = (\lambda_1 \geq \cdots \geq \lambda_s)$ be a partition
of $k$. We have
\begin{equation}
\Hilb(R_{n,\lambda}; q) = \sum_{\sigma \in \OP_{n,\lambda}} q^{\coinv(\sigma)}.
\end{equation}
\end{cor}








\acknowledgements{The authors are grateful to Sean Griffin and Andy Wilson for helpful conversations.
This research took place as a Research Experience for Undergraduates project at UC San Diego
in 2019-2020.}

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