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%% define your title in the usual way
\title{The Petrie symmetric functions}

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\author{Darij Grinberg\thanks{\href{mailto:darijgrinberg@gmail.com}{darijgrinberg@gmail.com}.}\addressmark{1}}

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\address{\addressmark{1}Mathematics Department, Drexel University, Philadelphia, PA, USA}

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\abstract{For any positive integer $k$ and nonnegative integer $m$, we consider the
symmetric function $G\left(  k,m\right)  $ defined as the sum of all monomials
of degree $m$ that contain no exponents larger than $k-1$. We call $G\left(
k,m\right)  $ a \textit{Petrie symmetric function} in honor of Flinders
Petrie, as the coefficients in its expansion in the Schur basis are
determinants of Petrie matrices (and thus belong to $\left\{ -1,0,1\right\}
$ by a classical result of Gordon and Wilkinson). More generally, we prove a
Pieri-like rule for expanding a product of the form $G\left(  k,m\right)
\cdot s_{\mu}$ in the Schur basis whenever $\mu$ is a partition; all
coefficients in this expansion belong to $\left\{ -1,0,1\right\}  $. We show
a further formula for $G\left(  k,m\right)  $ and prove that $G\left(
k,1\right)  ,G\left(  k,2\right)  ,G\left(  k,3\right)  ,\ldots$ form an
algebraically independent generating set for the symmetric functions when
$1-k$ is invertible in the base ring. We prove a conjecture of Liu and Polo
about the expansion of $G\left(  k,2k-1\right)
$ in the Schur basis.
We then take our Pieri-like rule as an impetus to pose a different question:
What other symmetric functions $f$ have the property that each product
$f s_\mu$ expands in the Schur basis with all coefficients belonging to
$\left\{-1,0,1\right\}$? We call this property \textit{MNability} due to its
most classical instance (besides the Pieri rules, which don't use $-1$
coefficients) being the Murnaghan--Nakayama rule.
Surprisingly, we find a number of infinite families of MNable symmetric
functions besides the classical ones.}

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\keywords{symmetric functions, Petrie matrices, Murnaghan--Nakayama rule, Pieri rules, Schur functions, determinants}

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\begin{document}

\maketitle

\section{Introduction}

In the course of computing the cohomology of a line bundle in characteristic
$p$, Liu and Polo \cite{LiuPol19} have encountered a symmetric function that
can be defined as the sum of all monomials of degree $2p-1$ that contain no
exponents larger than $p-1$. Using representation theory, they found a simple
expansion of this function in the Schur basis \cite[Corollary 1.4.4]%
{LiuPol19}, which prompted them to ask whether this expansion also holds for
non-prime $p$ (in which case their argument no longer applies).

Indeed, it does (see \cref{sect.liu-polo} below). From a combinatorial
point of view, it is natural to study an even more general family of symmetric
functions: We fix a commutative ring $\mathbf{k}$. For any integers $k\geq1$
and $m\geq0$, we let $G\left(  k,m\right)  $ be the sum of all monomials (in
$x_{1},x_{2},x_{3},\ldots$) of degree $m$ that contain no exponents larger
than $k-1$. This $G\left(  k,m\right)  $ is a symmetric function; moreover, if
it is expanded in the Schur basis of the ring of symmetric functions, then the
coefficients can be expressed as determinants of \emph{Petrie matrices} (i.e.,
matrices with each column having the form $\left(  0,0,\ldots,0,1,1,\ldots
,1,0,0,\ldots,0\right)  ^{T}$). Thus, we call $G\left(  k,m\right)  $ a
\emph{Petrie function} in honor of Flinders Petrie, whose invention of
contextual seriation gave birth to the notion of Petrie matrices. By a result
of Gordon and Wilkinson \cite{GorWil74}, Petrie matrices are unimodular; thus,
the coefficients in the Schur expansion of $G\left(  k,m\right)  $ belong to
$\left\{  -1,0,1\right\}  $.

More generally, if $\mu$ is any partition, then we can expand the product
$G\left(  k,m\right)  \cdot s_{\mu}$ in the Schur basis; all coefficients in
this Pieri-like expansion are determinants of Petrie matrices as well (and
thus belong to $\left\{  -1,0,1\right\}  $). At least for
$\mu=\varnothing$, the coefficients have a combinatorial interpretation.

We show a further formula for $G\left(  k,m\right)  $ in terms of Frobenius
homomorphisms $\mathbf{f}_{n}$ (also known as plethysm by the power-sum
function $p_{n}$), and we use it to show that $G\left(  k,1\right)  ,G\left(
k,2\right)  ,G\left(  k,3\right)  ,\ldots$ form an algebraically independent
generating set for the symmetric functions when $1-k$ is invertible in
$\mathbf{k}$.

We then revisit our expansion of $G\left(  k,m\right)  \cdot s_{\mu}$ to ask a
more general question (\cref{sect.MNable}): What other symmetric
functions $f$ have the property that each product $fs_{\mu}$ expands in the
Schur basis with all coefficients belonging to $\left\{  -1,0,1\right\}  $ ?
We call such $f$ \emph{MNable}; examples of MNable symmetric functions are the
classical functions $h_{m},e_{m},p_{m}$ (by the Pieri and the
Murnaghan--Nakayama rule, the latter of which gave MNability its name) and the
Petrie functions $G\left(  k,m\right)  $ (by the above). Surprisingly, we have
found several other MNable symmetric functions, such as the products
$p_{i}p_{j}$ with $i\neq j$, or the differences $h_{m}-e_{m}$, $h_{m}-p_{m}$ and
$h_{m}-p_{m}-e_{m}$ for even $m$.

Most results in this abstract are proved in the draft \cite{article}.

Some results below
(in particular, \cref{thm.G.main,thm.petk.explicit} in an equivalent form)
have been independently found by H. Fu and Z. Mei
\cite{FuMei20}.

\section{Definitions}

Our notations follow \cite[Chapter 2]{GriRei}. We let $\mathbb{N}=\left\{
0,1,2,\ldots\right\}  $.

We fix a commutative ring $\mathbf{k}$. We let $\Lambda$ denote the ring of
symmetric functions (i.e., symmetric power series of bounded degree) in
infinitely many variables $x_{1},x_{2},x_{3},\ldots$ over $\mathbf{k}$. This
is a $\mathbf{k}$-subalgebra of the $\mathbf{k}$-algebra $\mathbf{k}\left[
\left[  x_{1},x_{2},x_{3},\ldots\right]  \right]  $ of formal power series.

A \emph{weak composition} means an infinite sequence $\left(  \alpha
_{1},\alpha_{2},\alpha_{3},\ldots\right)  $ of nonnegative integers such that
only finitely many $i$ satisfy $\alpha_{i}\neq0$. If $\alpha$ is a weak
composition, then $\alpha_{i}$ is the $i$-th entry of $\alpha$ (so
that $\alpha=\left(  \alpha_{1},\alpha_{2},\alpha_{3},\ldots\right)  $), and
$\left\vert \alpha\right\vert $ is the sum $\alpha_{1}+\alpha
_{2}+\alpha_{3}+\cdots\in\mathbb{N}$ (and is called the \emph{size} of
$\alpha$). We let $\operatorname*{WC}$ denote the set of all weak compositions.

For any weak composition $\alpha$, we let $\mathbf{x}^{\alpha}$ denote the
monomial $x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}x_{3}^{\alpha_{3}}\cdots$. These
monomials $\mathbf{x}^{\alpha}$ are all the monomials in $\mathbf{k}\left[
\left[  x_{1},x_{2},x_{3},\ldots\right]  \right]  $.

A weak composition $\alpha$ will be identified with the $\ell$-tuple $\left(
\alpha_{1},\alpha_{2},\ldots,\alpha_{\ell}\right)  $ whenever $\ell
\in\mathbb{N}$ satisfies $\alpha_{\ell+1}=\alpha_{\ell+2}=\alpha_{\ell
+3}=\cdots=0$.

A \emph{partition} means a weak composition $\alpha$ such that $\alpha_{1}%
\geq\alpha_{2}\geq\alpha_{3}\geq\cdots$. We let $\operatorname*{Par}$ denote
the set of all partitions. For each $n\in\mathbb{N}$, we let
$\operatorname*{Par}\nolimits_{n}$ denote the set of all partitions
$\alpha$ satisfying $\left|\alpha\right| = n$.

The $\mathbf{k}$-module $\Lambda$ has several bases indexed by the set
$\operatorname*{Par}$. The simplest one is the \emph{monomial basis} $\left(
m_{\lambda}\right)  _{\lambda\in\operatorname*{Par}}$, whose elements
$m_{\lambda}$ are the sums of the orbits of the monomials $\mathbf{x}%
^{\alpha}$ under the \textquotedblleft permutation of
variables\textquotedblright\ action of the infinite symmetric group. More
precisely, for any partition $\lambda$, we can define the \emph{monomial
symmetric function }$m_{\lambda}\in\Lambda$ by
\[
m_{\lambda}=\sum\mathbf{x}^{\alpha},
\]
where the sum ranges over all weak compositions $\alpha\in\operatorname*{WC}$
that can be obtained from $\lambda$ by permuting entries. For example,
\[
m_{\left(  2,2,1\right)  }=\sum_{i<j<k}x_{i}^{2}x_{j}^{2}x_{k}+\sum
_{i<j<k}x_{i}^{2}x_{j}x_{k}^{2}+\sum_{i<j<k}x_{i}x_{j}^{2}x_{k}^{2}.
\]
As $\lambda$ ranges over all of $\operatorname*{Par}$, the symmetric functions
$m_{\lambda}$ form a basis of the $\mathbf{k}$-module $\Lambda$.

Other prominent symmetric functions in $\Lambda$ are:

\begin{itemize}
\item the \emph{complete homogeneous symmetric functions }$h_{n}$ defined for
all $n\in\mathbb{Z}$ by%
\[
h_{n}=\sum_{i_{1}\leq i_{2}\leq\cdots\leq i_{n}}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{n}}=\sum_{\substack{\alpha\in\operatorname*{WC};\\\left\vert
\alpha\right\vert =n}}\mathbf{x}^{\alpha}=\sum_{\lambda\in\operatorname*{Par}%
\nolimits_{n}}m_{\lambda}.
\]
(Thus, $h_{0}=1$ and $h_{n}=0$ for all $n<0$.)

\item the \emph{elementary symmetric functions }$e_{n}$ defined for all
$n\in\mathbb{Z}$ by%
\[
e_{n}=\sum_{i_{1}<i_{2}<\cdots<i_{n}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}%
=\sum_{\substack{\alpha\in\operatorname*{WC}\cap\left\{  0,1\right\}
^{\infty};\\\left\vert \alpha\right\vert =n}}\mathbf{x}^{\alpha}.
\]
(Thus, $e_{0}=1$ and $e_{n}=0$ for all $n<0$. If $n>0$, then $e_{n}=m_{\left(
1,1,\ldots,1\right)  }$, where $\left(  1,1,\ldots,1\right)  $ is an $n$-tuple.)

\item the \emph{power-sum symmetric functions} $p_{n}$ defined for all
positive integers $n$ by%
\[
p_{n}=x_{1}^{n}+x_{2}^{n}+x_{3}^{n}+\cdots=m_{\left(  n\right)  }.
\]

\end{itemize}

But most remarkable of all are the \emph{Schur functions }$s_{\lambda}$ for
$\lambda\in\operatorname*{Par}$. One way to define the Schur function
$s_{\lambda}$ corresponding to a partition $\lambda$ is as follows:
\[
s_{\lambda}=\sum\mathbf{x}_{T},
\]
where the sum ranges over all semistandard tableaux $T$ of shape $\lambda$,
and where $\mathbf{x}_{T}$ denotes the monomial obtained by multiplying the
$x_{i}$ for all entries $i$ of $T$. The fact that $s_{\lambda}\in\Lambda$ is
nontrivial (see, e.g., \cite[Proposition 2.2.4]{GriRei}); the $s_{\lambda}$ are rich
in interesting and nontrivial properties (\cite[Chapter 2]{GriRei}, \cite[Chapter
7]{Stanley-EC2}, etc.). In particular, the family $\left(  s_{\lambda}\right)
_{\lambda\in\operatorname*{Par}}$ is a basis of the $\mathbf{k}$-module
$\Lambda$, known as the \emph{Schur basis}.

\section{Definition of the Petrie functions}

We are now ready to define the functions we will study:

\begin{definition}
\begin{enumerate}[label={\bf (\alph*)},wide, labelwidth=!, labelindent=0pt, itemsep=-4pt]
\item For any positive integer $k$, we let%
\[
G\left(  k\right)  =\sum_{\substack{\alpha\in\operatorname*{WC};\\\alpha
_{i}<k\text{ for all }i}}\mathbf{x}^{\alpha}.
\]
This is a symmetric formal power series in $\mathbf{k}\left[  \left[
x_{1},x_{2},x_{3},\ldots\right]  \right]  $ (but does not lie in $\Lambda$
in general, since it contains monomials of arbitrarily high degrees).

\item For any positive integer $k$ and any $m\in\mathbb{N}$, we let%
\[
G\left(  k,m\right)  =\sum_{\substack{\alpha\in\operatorname*{WC};\\\left\vert
\alpha\right\vert =m;\\\alpha_{i}<k\text{ for all }i}}\mathbf{x}^{\alpha}%
\in\Lambda.
\]
\end{enumerate}
\end{definition}

For example,%
\begin{align*}
G\left(  3,4\right)   &  =\sum_{i<j<k<\ell}x_{i}x_{j}x_{k}x_{\ell}%
+\sum_{i<j<k}x_{i}^{2}x_{j}x_{k}+\sum_{i<j<k}x_{i}x_{j}^{2}x_{k}+\sum
_{i<j<k}x_{i}x_{j}x_{k}^{2}+\sum_{i<j}x_{i}^{2}x_{j}^{2}\\
&  =m_{\left(  1,1,1,1\right)  }+m_{\left(  2,1,1\right)  }+m_{\left(
2,2\right)  }.
\end{align*}


We suggest to name $G\left(  k\right)  $ and $G\left(  k,m\right)  $ the
\emph{Petrie functions}, for reasons that \cref{thm.G.main} and
\cref{cor.Gkm.main} will elucidate. We begin with some easy facts:

\begin{proposition}
\label{prop.G.basics1}Let $k$ be a positive integer.
Then,
\[
G\left(  k\right)  =\sum_{\substack{\alpha\in\operatorname*{WC};\\\alpha
_{i}<k\text{ for all }i}}\mathbf{x}^{\alpha}=\sum_{\substack{\lambda
\in\operatorname*{Par};\\\lambda_{i}<k\text{ for all }i}}m_{\lambda}%
=\prod_{i=1}^{\infty}\left(  x_{i}^{0}+x_{i}^{1}+\cdots+x_{i}^{k-1}\right)  .
\]
\end{proposition}

\begin{proposition}
\label{prop.G.basics2}Let $k$ be a positive integer.
Let $m \in \mathbb{N}$.
\begin{enumerate}[label={\bf (\alph*)},wide, labelwidth=!, labelindent=0pt, itemsep=-4pt]
\item The symmetric function $G\left(  k,m\right)  $ is the $m$-th
degree homogeneous component of $G\left(  k\right)  $.
\item We have
\[
G\left(  k,m\right)  =\sum_{\substack{\alpha\in\operatorname*{WC};\\\left\vert
\alpha\right\vert =m;\\\alpha_{i}<k\text{ for all }i}}\mathbf{x}^{\alpha}%
=\sum_{\substack{\lambda\in\operatorname*{Par};\\\left\vert \lambda\right\vert
=m;\\\lambda_{i}<k\text{ for all }i}}m_{\lambda}.
\]
\item If $k>m$, then $G\left(  k,m\right)  =h_{m}$.
\item If $k=2$, then $G\left(  k,m\right)  =e_{m}$.
\end{enumerate}
\end{proposition}

Parts \textbf{(c)} and \textbf{(d)} of \cref{prop.G.basics2}
show that the Petrie
functions $G\left(  k,m\right)  $ can be seen as interpolating between the
$h_{m}$ and the $e_{m}$. Another easily established identity is $G\left(
m,m\right)  =h_{m}-p_{m}$ for each positive integer $m$.

It is also not hard to see that the comultiplication $\Delta$ of the
Hopf algebra $\Lambda$ (see \cite[Section 2.1]{GriRei} for its
definition) satisfies
\[
\Delta\left(G\left(k, m\right)\right)
= \sum_{i=0}^m G\left(k, i\right) \otimes G\left(k, m-i\right)
\]
for each $k > 0$ and $m \in \NN$.

The Petrie function $G\left(  3\right)  $ has appeared in \cite[Exercise
7.3]{Stanley-EC2}, where it was expanded as a polynomial in $e_{1},e_{2}%
,e_{3},\ldots$ (a result of Gessel). We shall now expand $G\left(  k\right)  $
and $G\left(  k,m\right)  $ in terms of Schur functions. For this, we need to
define some notations.

\section{The Schur expansions of $G\left(  k\right)  $ and $G\left(
k,m\right)  $}

If $\mathcal{A}$ is any logical statement, then $\left[  \mathcal{A}\right]  $
shall denote the \emph{truth value} of $\mathcal{A}$ (that is, $1$ if
$\mathcal{A}$ is true, and $0$ if $\mathcal{A}$ is false). We use the notation
$\left(  a_{i,j}\right)  _{1\leq i\leq\ell,\ 1\leq j\leq\ell}$ for the
$\ell\times\ell$-matrix whose $\left(  i,j\right)  $-th entry is $a_{i,j}$ for
each $i,j\in\left\{  1,2,\ldots,\ell\right\}  $.

\begin{definition}
Let $\lambda=\left(  \lambda_{1},\lambda_{2},\ldots,\lambda_{\ell}\right)
\in\operatorname*{Par}$, and let $k$ be a positive integer. Then, the
$k$\emph{-Petrie number} $\operatorname*{pet}\nolimits_{k}\left(
\lambda\right)  $ of $\lambda$ is the integer defined by%
\[
\operatorname*{pet}\nolimits_{k}\left(  \lambda\right)  =\det\left(  \left(
\left[  0\leq\lambda_{i}-i+j<k\right]  \right)  _{1\leq i\leq\ell,\ 1\leq
j\leq\ell}\right)  .
\]
Note that this integer does not depend on the choice of $\ell$ (in the sense
that it does not change if we enlarge $\ell$ by adding trailing zeroes to the
representation of $\lambda$).
\end{definition}

\begin{example}
Let $\lambda$ be the partition $\left(  3,1,1\right)  \in\operatorname*{Par}$,
let $\ell=3$, and let $k$ be a positive integer. Then, the definition of
$\operatorname*{pet}\nolimits_{k}\left(  \lambda\right)  $ yields%
\begin{align*}
\operatorname*{pet}\nolimits_{k}\left(  \lambda\right)   &  =\det\left(
\begin{array}
[c]{ccc}%
\left[  0\leq\lambda_{1}<k\right]  & \left[  0\leq\lambda_{1}+1<k\right]  &
\left[  0\leq\lambda_{1}+2<k\right] \\
\left[  0\leq\lambda_{2}-1<k\right]  & \left[  0\leq\lambda_{2}<k\right]  &
\left[  0\leq\lambda_{2}+1<k\right] \\
\left[  0\leq\lambda_{3}-2<k\right]  & \left[  0\leq\lambda_{3}-1<k\right]  &
\left[  0\leq\lambda_{3}<k\right]
\end{array}
\right) \\
&  =\det\left(
\begin{array}
[c]{ccc}%
\left[  0\leq3<k\right]  & \left[  0\leq4<k\right]  & \left[  0\leq5<k\right]
\\
\left[  0\leq0<k\right]  & \left[  0\leq1<k\right]  & \left[  0\leq2<k\right]
\\
\left[  0\leq-1<k\right]  & \left[  0\leq0<k\right]  & \left[  0\leq
1<k\right]
\end{array}
\right)
\end{align*}
(since $\lambda_{1}=3$ and $\lambda_{2}=1$ and $\lambda_{3}=1$). Thus, taking
$k=4$, we obtain%
\[
\operatorname*{pet}\nolimits_{4}\left(  \lambda\right)  =\det\left(
\begin{array}
[c]{ccc}%
\left[  0\leq3<4\right]  & \left[  0\leq4<4\right]  & \left[  0\leq5<4\right]
\\
\left[  0\leq0<4\right]  & \left[  0\leq1<4\right]  & \left[  0\leq2<4\right]
\\
\left[  0\leq-1<4\right]  & \left[  0\leq0<4\right]  & \left[  0\leq
1<4\right]
\end{array}
\right)  =\det\left(
\begin{array}
[c]{ccc}%
1 & 0 & 0\\
1 & 1 & 1\\
0 & 1 & 1
\end{array}
\right)  =0.
\]
On the other hand, taking $k=5$, we obtain%
\[
\operatorname*{pet}\nolimits_{5}\left(  \lambda\right)  =\det\left(
\begin{array}
[c]{ccc}%
\left[  0\leq3<5\right]  & \left[  0\leq4<5\right]  & \left[  0\leq5<5\right]
\\
\left[  0\leq0<5\right]  & \left[  0\leq1<5\right]  & \left[  0\leq2<5\right]
\\
\left[  0\leq-1<5\right]  & \left[  0\leq0<5\right]  & \left[  0\leq
1<5\right]
\end{array}
\right)  =\det\left(
\begin{array}
[c]{ccc}%
1 & 1 & 0\\
1 & 1 & 1\\
0 & 1 & 1
\end{array}
\right)  =-1.
\]

\end{example}

\begin{proposition}
\label{prop.petk.-101}Let $\lambda\in\operatorname*{Par}$, and let $k$ be a
positive integer. Then, $\operatorname*{pet}\nolimits_{k}\left(
\lambda\right)  \in\left\{  -1,0,1\right\}  $.
\end{proposition}

\begin{proof}
[Proof sketch.]We will use the concept of Petrie matrices (see \cite[Theorem
1]{GorWil74}). Each row of the matrix $\left(  \left[  0\leq\lambda
_{i}-i+j<k\right]  \right)  _{1\leq i\leq\ell,\ 1\leq j\leq\ell}$ has the
form
\[
(\underbrace{0,0,\ldots,0}_{a\text{ zeroes}},\underbrace{1,1,\ldots
,1}_{b\text{ ones}},\underbrace{0,0,\ldots,0}_{c\text{ zeroes}}%
)\ \ \ \ \ \ \ \text{for some $a,b,c\in\mathbb{N}$
(where any of $a,b,c$ can be $0$)}.
\]
Thus, the matrix $\left(  \left[
0\leq\lambda_{i}-i+j<k\right]  \right)  _{1\leq i\leq\ell,\ 1\leq j\leq\ell}$
is the transpose of a Petrie matrix. Hence, its determinant belongs to
$\left\{ -1,0,1\right\}  $ (since \cite[Theorem 1]{GorWil74} shows that the
determinant of any square Petrie matrix belongs to
$\left\{  -1,0,1\right\}  $).
\end{proof}

We can now expand the Petrie symmetric functions $G\left(  k\right)  $ in the
basis $\left(  s_{\lambda}\right)  _{\lambda\in\operatorname*{Par}}$ of
$\Lambda$:

\begin{theorem}
\label{thm.G.main}Let $k$ be a positive integer. Then,%
\[
G\left(  k\right)  =\sum_{\lambda\in\operatorname*{Par}}\operatorname*{pet}%
\nolimits_{k}\left(  \lambda\right)  s_{\lambda}.
\]

\end{theorem}

\begin{corollary}
\label{cor.Gkm.main}Let $k$ be a positive integer. Let $m\in\mathbb{N}$. Then,%
\[
G\left(  k,m\right)  =\sum_{\lambda\in\operatorname*{Par}\nolimits_{m}%
}\operatorname*{pet}\nolimits_{k}\left(  \lambda\right)  s_{\lambda}.
\]

\end{corollary}

These two results are particular cases of more general facts stated
below (\cref{thm.G.pieri} and \cref{cor.Gkm.pieri}).

The $k$-Petrie numbers can be described more explicitly:

\begin{theorem}
\label{thm.petk.explicit}Let $\lambda\in\operatorname*{Par}$, and let $k$ be a
positive integer. Let $\mu=\lambda^{t}$ be the \emph{conjugate}
of $\lambda$ (that is, the partition $\mu$ defined by setting $\mu_i
= \left|\left\{ j \geq 1 \mid \lambda_j \geq i \right\}\right|$ for all $i$).
\begin{enumerate}[label={\bf (\alph*)},wide, labelwidth=!, labelindent=0pt, itemsep=-4pt]
\item If $\mu_{k}\neq0$, then $\operatorname*{pet}\nolimits_{k}\left(
\lambda\right)  =0$. From now on, let us assume $\mu_{k}=0$. For each $i \in \{1,2,\linebreak[0]\ldots,k-1\}$,
let $\gamma_i$ be the unique element of
$\left\{1,2,\ldots,k\right\}$ that is congruent to $\mu_i - i$
modulo $k$.

\item If the $k-1$ numbers $\gamma_{1},\gamma_{2},\ldots,\gamma_{k-1}$
are not distinct, then $\operatorname*{pet}\nolimits_{k}\left(  \lambda
\right)  =0$.

\item If the $k-1$ numbers $\gamma_{1},\gamma_{2}%
,\ldots,\gamma_{k-1}$ are distinct, then $\operatorname*{pet}\nolimits_{k}\left(  \lambda
\right)$ is a certain power of $-1$ (see \cite{article} for details).
\end{enumerate}
\end{theorem}


\section{\label{sect.pieri}A \textquotedblleft Pieri\textquotedblright\ rule}

It turns out that \cref{thm.G.main} can be generalized. For that, we
need to define a \textquotedblleft relative\textquotedblright\ version of
Petrie numbers:

\begin{definition}
Let $\lambda=\left(  \lambda_{1},\lambda_{2},\ldots,\lambda_{\ell}\right)
\in\operatorname*{Par}$ and $\mu=\left(  \mu_{1},\mu_{2},\ldots,\mu_{\ell
}\right)  \in\operatorname*{Par}$, and let $k$ be a positive integer. Then,
the $k$\textit{-Petrie number} $\operatorname*{pet}\nolimits_{k}\left(
\lambda,\mu\right)  $ of $\lambda$ and $\mu$ is the integer defined by%
\[
\operatorname*{pet}\nolimits_{k}\left(  \lambda,\mu\right)  =\det\left(
\left(  \left[  0\leq\lambda_{i}-\mu_{j}-i+j<k\right]  \right)  _{1\leq
i\leq\ell,\ 1\leq j\leq\ell}\right)  .
\]
Note that this integer does not depend on the choice of $\ell$ (in the sense
that it does not change if we enlarge $\ell$ by adding trailing zeroes to the
representations of $\lambda$ and $\mu$).
\end{definition}

The following proposition generalizes (and is proved similarly to) \cref{prop.petk.-101}:

\begin{proposition}
\label{prop.petkrel.-101}Let $\lambda\in\operatorname*{Par}$ and $\mu
\in\operatorname*{Par}$, and let $k$ be a positive integer. Then,
$\operatorname*{pet}\nolimits_{k}\left(  \lambda,\mu\right)  \in\left\{
-1,0,1\right\}  $.
\end{proposition}

Now, we have the following generalizations of \cref{thm.G.main} and
\cref{cor.Gkm.main}:

\begin{theorem}
\label{thm.G.pieri}Let $k$ be a positive integer. Let $\mu\in
\operatorname*{Par}$. Then,%
\[
G\left(  k\right)  \cdot s_{\mu}=\sum_{\lambda\in\operatorname*{Par}%
}\operatorname*{pet}\nolimits_{k}\left(  \lambda,\mu\right)  s_{\lambda}.
\]

\end{theorem}

\begin{corollary}
\label{cor.Gkm.pieri}Let $k$ be a positive integer. Let $m\in\mathbb{N}$. Let
$\mu\in\operatorname*{Par}$. Then,%
\[
G\left(  k,m\right)  \cdot s_{\mu}=\sum_{\lambda\in\operatorname*{Par}%
\nolimits_{m+\left\vert \mu\right\vert }}\operatorname*{pet}\nolimits_{k}%
\left(  \lambda,\mu\right)  s_{\lambda}.
\]

\end{corollary}

We have two proofs of \cref{thm.G.pieri}: one using the skew Cauchy and
the Jacobi--Trudi identities, and another using the approach to Schur
polynomials via alternants. (See \cite{article} for the second proof.)
\cref{cor.Gkm.pieri} easily follows from \cref{thm.G.pieri}.

We are not aware of any combinatorial rules for $\operatorname*{pet}%
\nolimits_{k}\left(  \lambda,\mu\right)  $ other than the (general, but
recursive and rather indirect) algorithmic description given in
\cite{GorWil74} for determinants of arbitrary square Petrie matrices.

\section{\label{sect.frob}The Frobenius formula and Petrie generating sets}

We shall next state another formula for the Petrie symmetric functions
$G\left(  k,m\right)  $. For this formula, we need the following definition
(\cite[Exercise 2.9.9]{GriRei}):

\begin{definition}
\label{def.frob}Let $n$ be a positive integer. We define a map $\mathbf{f}%
_{n}:\Lambda\rightarrow\Lambda$ by setting%
\[
\mathbf{f}_{n}\left(  a\right)  =a\left(  x_{1}^{n},x_{2}^{n},x_{3}^{n}%
,\ldots\right)  \ \ \ \ \ \ \ \ \ \ \text{for each }a\in\Lambda.
\]
This map $\mathbf{f}_{n}$ is called the $n$\emph{-th Frobenius endomorphism}
of $\Lambda$.
\end{definition}

It is known (e.g., \cite[Exercise 2.9.9(d)]{GriRei}) that this map
$\mathbf{f}_{n}:\Lambda\rightarrow\Lambda$ is a $\mathbf{k}$-algebra
endomorphism of $\Lambda$ (and even a Hopf algebra endomorphism, using the
appropriate Hopf structure). In terms of plethysm (\cite[Ch. 7, Definition
A.2.6]{Stanley-EC2}), it is simply described by $\mathbf{f}_{n}\left(
a\right)  =a\left[  p_{n}\right]  $ for each $a\in\Lambda$ (and also by
$\mathbf{f}_{n}\left(  a\right)  =p_{n}\left[  a\right]  $ if $\mathbf{k}%
=\mathbb{Z}$).

We now have a new formula for $G\left(  k,m\right)  $:

\begin{theorem}
\label{thm.G.frob}Let $k$ be a positive integer. Let $m\in\mathbb{N}$. Then,%
\[
G\left(  k,m\right)  =\sum_{i\in\mathbb{N}}\left(  -1\right)  ^{i}%
h_{m-ki}\cdot\mathbf{f}_{k}\left(  e_{i}\right)  .
\]
(The sum on the right hand side of this equality is well-defined, since all
sufficiently large $i\in\mathbb{N}$ satisfy $m-ki<0$ and thus $h_{m-ki}=0$.)
\end{theorem}

This theorem can be proved by an inclusion-exclusion-like computation or using
generating functions (the latter proof is given in \cite{article}).

\cref{thm.G.frob} can be used to derive the following:

\begin{theorem}
\label{thm.Gkm-genset}Fix a positive integer $k$. Assume that $1-k$ is
invertible in $\mathbf{k}$. Then, the family $\left(  G\left(  k,m\right)
\right)  _{m\geq1}=\left(  G\left(  k,1\right)  ,G\left(  k,2\right)
,G\left(  k,3\right)  ,\ldots\right)  $ is an algebraically independent
generating set of the commutative $\mathbf{k}$-algebra $\Lambda$.
\end{theorem}

Thus, products of several elements of this family form a basis of $\Lambda$
(if $1-k$ is invertible in $\mathbf{k}$). These bases remain to be studied.

\section{\label{sect.liu-polo}The Liu--Polo conjecture}

We now sketch the answer to the question posed in \cite[Remark 1.4.5]%
{LiuPol19} by Liu and Polo. By studying cohomology in positive characteristic,
they have proved the following identity for all prime numbers $n$:

\begin{theorem}
\label{thm.claim}Let $n$ be an integer such that $n>1$. Then,%
\begin{equation}
\sum_{\substack{\lambda\in\operatorname*{Par}\nolimits_{2n-1};\\\left(
n-1,n-1,1\right)  \triangleright\lambda}}m_{\lambda}=\sum_{i=0}^{n-2}\left(
-1\right)  ^{i}s_{\left(  n-1,n-1-i,1^{i+1}\right)  }.
\label{eq.thm.claim.claim}%
\end{equation}

\end{theorem}

Here, the symbol $\triangleright$ stands for \emph{dominance} of partitions
(also known as majorization); i.e., for two partitions $\lambda$ and $\mu$
satisfying $\left|\lambda\right| = \left|\mu\right|$, we have
\[
\lambda\triangleright\mu\ \ \ \ \ \ \ \ \ \ \text{if and only if}%
\ \ \ \ \ \ \ \ \ \ \left(  \lambda_{1}+\lambda_{2}+\cdots+\lambda_{i}\geq
\mu_{1}+\mu_{2}+\cdots+\mu_{i}\text{ for all }i\right)  .
\]
Also, the power-like notation \textquotedblleft$1^{i+1}$\textquotedblright\ in
the partition on the right hand side of \eqref{eq.thm.claim.claim}
stands for a sequence of $i+1$ entries
all equal to $1$. Thus, $\left(  n-1,n-1-i,1^{i+1}\right)
=(n-1,n-1-i,\underbrace{1,1,\ldots,1}_{i+1\text{ times}})$.

We can prove \cref{thm.claim} for all $n$ as follows. The first step is
to recognize that the left hand side of \eqref{eq.thm.claim.claim} is
$G\left(  n,2n-1\right)  $, because the partitions $\lambda\in
\operatorname*{Par}\nolimits_{2n-1}$ satisfying $\left(  n-1,n-1,1\right)
\triangleright\lambda$ are precisely the partitions $\lambda\in
\operatorname*{Par}\nolimits_{2n-1}$ satisfying $\lambda_{i}<n$ for all $i$.
\cref{thm.G.main} gives an expansion of $G\left(  n,2n-1\right)  $ in
the Schur basis, if we content ourselves with knowing that the coefficients
are $n$-Petrie numbers. However, we want to know their exact values in order
to prove \eqref{eq.thm.claim.claim}. Thus, we proceed differently. An
application of \cref{thm.G.frob} (or a simple combinatorial argument)
yields
\[
G\left(  n,n+k\right)  =h_{n+k}-h_{k}p_{n}\ \ \ \ \ \ \ \ \ \ \text{for each
}k\in\left\{  0,1,\ldots,n-1\right\}  .
\]
Thus, in particular, $G\left(  n,2n-1\right)  =h_{2n-1}-h_{n-1}p_{n}$.

Now, we recall the \emph{skewing operations} $f^{\perp}:\Lambda\rightarrow
\Lambda$ for all $f\in\Lambda$ as defined in \cite[Section 2.8]{GriRei} (and in
various other places). All we need to know about them is that for each
$i\in\mathbb{N}$, the skewing operation $e_{i}^{\perp}:\Lambda\rightarrow
\Lambda$ is the $\mathbf{k}$-linear map that sends each Schur function
$s_{\lambda}$ to the skew Schur function $s_{\lambda/\left(  1^i\right)  }$.

For any $m\in\mathbb{N}$, we define a map $\mathbf{B}_{m}:\Lambda
\rightarrow\Lambda$ by setting%
\[
\mathbf{B}_{m}\left(  f\right)  =\sum_{i\in\mathbb{N}}\left(  -1\right)
^{i}h_{m+i}e_{i}^{\perp}f\ \ \ \ \ \ \ \ \ \ \text{for all }f\in
\Lambda\text{.}%
\]
It is known (\cite[Exercise 2.9.1(a)]{GriRei}) that this map $\mathbf{B}_{m}$
is well-defined and $\mathbf{k}$-linear. Moreover, \cite[Exercise
2.9.1(b)]{GriRei} shows that if $\lambda\in\operatorname{Par}$ and
$m\in\mathbb{Z}$ satisfy $m\geq\lambda_{1}$, then%
\begin{equation}
\mathbf{B}_{m}\left(  s_{\lambda}\right)  =s_{\left(  m,\lambda_{1}%
,\lambda_{2},\lambda_{3},\ldots\right)  }. \label{eq.schur-row-adder-2}%
\end{equation}
(This map $\mathbf{B}_{m}$ is known as the \emph{$m$-th Bernstein operator}
\cite[Section 4.20(a)]{Zelevi81} or --- in honor of \eqref{eq.schur-row-adder-2} ---
a \emph{Schur row-adder} \cite{Garsia10}.) On the other hand, it is not hard
to see that%
\[
\mathbf{B}_{m}\left(  h_{n}\right)  =h_{m}h_{n}-h_{m+1}h_{n-1}%
\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ \mathbf{B}_{m}\left(
p_{n}\right)  =h_{m}p_{n}-h_{m+n}%
\]
for each positive integer $n$ and each $m\in\left\{  0,1,\ldots,n\right\}  $.
Using these two equalities, we readily see that%
\begin{equation}
\mathbf{B}_{n-1}\left(  h_{n}-p_{n}\right)  =h_{2n-1}-h_{n-1}p_{n}=G\left(
n,2n-1\right)  . \label{pf.liu-polo.14}%
\end{equation}
On the other hand, \cite[Exercise 5.4.12(g)]{GriRei} (or the
Murnaghan--Nakayama rule) yields%
\[
p_{n}=\sum_{i=0}^{n-1}\left(  -1\right)  ^{i}s_{\left(  n-i,1^{i}\right)  }.
\]
Subtracting this from $h_{n}=s_{\left(  n\right)  }=s_{\left(  n-0,1^{0}%
\right)  }$, we find
\[
h_{n}-p_{n}=\sum_{i=0}^{n-2}\left(  -1\right)  ^{i}s_{\left(  n-1-i,1^{i+1}%
\right)  }.
\]
Applying the map $\mathbf{B}_{n-1}$ to this equality, we obtain%
\[
\mathbf{B}_{n-1}\left(  h_{n}-p_{n}\right)  =\sum_{i=0}^{n-2}\left(
-1\right)  ^{i}\mathbf{B}_{n-1}\left(  s_{\left(  n-1-i,1^{i+1}\right)
}\right)  =\sum_{i=0}^{n-2}\left(  -1\right)  ^{i}s_{\left(  n-1,n-1-i,1^{i+1}%
\right)  }%
\]
(by \eqref{eq.schur-row-adder-2}). Comparing this with \eqref{pf.liu-polo.14},
we find
\[
G\left(  n,2n-1\right)
= \sum\limits_{i=0}^{n-2}\left(  -1\right)  ^{i}s_{\left(
n-1,n-1-i,1^{i+1}\right)  } .
\]
Since the left hand side of \eqref{eq.thm.claim.claim} is $G\left(
n,2n-1\right)  $, we have thus proved \cref{thm.claim}.

\section{\label{sect.MNable}MNable symmetric functions}

Let us now take \cref{cor.Gkm.pieri} as inspiration to identify a
property of some symmetric functions that appears to have been hitherto unstudied.

Let $\mathbf{k}=\mathbb{Z}$ throughout this section. We recall the Hall inner
product $\left(  \cdot,\cdot\right)  :\Lambda\times\Lambda\rightarrow
\mathbf{k}$; it is the unique $\mathbf{k}$-bilinear form on $\Lambda$ that
satisfies $\left(  s_{\lambda},s_{\mu}\right)  =\delta_{\lambda,\mu}$ for all
$\lambda,\mu\in\operatorname*{Par}$. (In other words, it is the unique
$\mathbf{k}$-bilinear form on $\Lambda$ that makes $\left(  s_{\lambda
}\right)  _{\lambda\in\operatorname*{Par}}$ into an orthonormal basis.) See
\cite[Definition 2.5.12]{GriRei} or \cite[Section 7.9]{Stanley-EC2} (where it is
denoted by $\left\langle \cdot,\cdot\right\rangle $) for its further properties.

\begin{definition}
\begin{enumerate}[label={\bf (\alph*)},wide, labelwidth=!, labelindent=0pt, itemsep=-4pt]
\item A symmetric function $f\in\Lambda$ will be called \emph{signed
multiplicity-free} if $f$ can be expanded as a linear combination of distinct
Schur functions with all coefficients in $\left\{  -1,0,1\right\}  $. (That
is, if the Hall inner product $\left(  f,s_{\mu}\right)  $ is $-1$, $0$ or
$1$ for each partition $\mu$.)

\item A symmetric function $f\in\Lambda$ will be called \emph{MNable}
if for each partition $\mu$, the product $fs_{\mu}$ is signed multiplicity-free.
\end{enumerate}
\end{definition}

For example, the symmetric function $h_{3}p_{2}$ is signed multiplicity-free,
since $h_{3}p_{2}=s_{\left(  5\right)  }+s_{\left(  3,2\right)  }-s_{\left(
3,1,1\right)  }$; but it is not MNable, since the product
\[
h_{3}p_{2}s_{\left(  2\right)  }=-s_{\left(  3,2,1,1\right)  }+s_{\left(
3,2,2\right)  }-s_{\left(  4,1,1,1\right)  }+s_{\left(  4,3\right)
}-s_{\left(  5,1,1\right)  }+2s_{\left(  5,2\right)  }+s_{\left(  6,1\right)
}+s_{\left(  7\right)  }%
\]
is not signed multiplicity-free (due to the coefficient of $s_{\left(
5,2\right)  }$ being $2$).

Roughly speaking, an $f\in\Lambda$ is MNable if and only if there is a
\textbf{M}urnaghan-\textbf{N}aka\-yama-like rule for multiplying Schur functions
by $f$. Thus, the name \textquotedblleft\textbf{MN}able\textquotedblright.

\begin{question}
\label{quest.MNable.which}Which symmetric functions are MNable?
\end{question}

It is not clear whether a full characterization of MNable symmetric functions
is even possible. However, there are many. Here is a non-exhaustive list:

\begin{theorem}
\label{thm.MNable.list}\begin{enumerate}[label={\bf (\alph*)},wide, labelwidth=!, labelindent=0pt, itemsep=-4pt]
\item The functions $h_{i}$ and $e_{i}$ are
MNable for each $i\in\mathbb{N}$.

\item The function $p_{i}$ is MNable for each positive integer $i$.

\item The Petrie function $G\left(  k,m\right)  $ and the difference
$G\left(  k,m\right)  -h_{m}$ are MNable for any integers $k\geq1$ and
$m\geq0$.

\item The differences $h_{i}-p_{i}$ and $h_{i}-e_{i}$ are MNable for
each positive integer $i$. (This includes $h_{1}-e_{1}=0$.)

\item  The difference $h_{i}-p_{i}-e_{i}$ is MNable for each
\textbf{even} positive integer $i$.

\item The product $p_{i}p_{j}$ is MNable whenever $i>j>0$.

\item The function $m_{\left(  k^{n}\right)  }$ as well as the
differences $h_{nk}-m_{(k^{n})}$ and $e_{nk}-\left(  -1\right)  ^{\left(
k-1\right)  n}m_{\left(  k^{n}\right)  }$ are MNable for any positive integers
$n$ and $k$ (where $\left(  k^{n}\right)  $ denotes the $n$-tuple $\left(
k,k,\ldots,k\right)  $).

\item If some $f\in\Lambda$ is MNable, then so are $-f$ and
$\omega\left(  f\right)  $, where $\omega:\Lambda\rightarrow\Lambda$ is the
\emph{fundamental involution} of $\Lambda$ (see \cite[Section 2.4]{GriRei} or
\cite[Section 7.6]{Stanley-EC2}).

\item A symmetric function $f\in\Lambda$ is MNable if and only if all
its homogeneous components are MNable.

\item If $f\in\Lambda$ is MNable and $k$ is a positive integer, then
$\mathbf{f}_{k}\left(  f\right)  $ is MNable. (See \cref{def.frob}
for the meaning of $\mathbf{f}_{k}$.)

\item A symmetric function $f\in\Lambda$ is MNable if and only if
$\left(  f,s_{\lambda/\mu}\right)  \in\left\{  -1,0,1\right\}  $ for each skew
partition $\lambda/\mu$.
\end{enumerate}
\end{theorem}

A few telegraphic remarks on the proofs are in order. Part \textbf{(a)} of
\cref{thm.MNable.list} follows from the Pieri and dual Pieri rules, as
part \textbf{(b)} does from the Murnaghan--Nakayama rule. The $G\left(
k,m\right)  $ claim in part \textbf{(c)} follows from \cref{cor.Gkm.pieri}; the $G\left(  k,m\right)  -h_{m}$ claim relies on the
fact that $\operatorname*{pet}\nolimits_{k}\left(  \lambda,\mu\right)
\in\left\{  0,1\right\}  $ if $\lambda/\mu$ is a horizontal strip. Parts
\textbf{(d)} and \textbf{(e)} can be shown by analyzing the rare cases in
which a skew partition can be two of \textquotedblleft horizontal
strip\textquotedblright, \textquotedblleft vertical strip\textquotedblright%
\ and \textquotedblleft rim hook\textquotedblright\ at once. Part \textbf{(f)}
follows from a study of rim hook tableaux. Part \textbf{(h)} follows from
the facts
that $\omega$ is an algebra automorphism and sends $s_\lambda$ to
$s_{\lambda^t}$. Part \textbf{(k)} is easy to see using skewing operators (or
simply using the fact that the same Littlewood--Richardson coefficients appear
in the formulas $s_{\mu}s_{\nu}=\sum_{\lambda\in\operatorname*{Par}}c_{\mu
,\nu}^{\lambda}s_{\lambda}$ and $s_{\lambda/\mu}=\sum_{\nu\in
\operatorname*{Par}}c_{\mu,\nu}^{\lambda}s_{\nu}$). Part \textbf{(i)} is easy.
Part \textbf{(j)} follows from part \textbf{(k)} and the SXP algorithm in
\cite{ChGaRe}. The $m_{\left(  k^{n}\right)  }$ claim in part \textbf{(g)}
follows from part \textbf{(j)} (since $m_{\left(  k^{n}\right)  }%
=\mathbf{f}_{k}\left(  e_{n}\right)  $); the rest of \textbf{(g)} follows by
studying skew partitions again.

Note that \cref{thm.MNable.list} \textbf{(k)} shows that the MNability
of a symmetric function can be tested in finite time: For each $d\in
\mathbb{N}$, there are only finitely many skew Schur functions $s_{\lambda
/\mu}$ of degree $d$.

The families in parts \textbf{(a)}--\textbf{(h)} and \textbf{(j)} of \cref{thm.MNable.list} cover all MNable homogeneous symmetric functions of
degree $<4$. In degree $4$, we have two further MNable symmetric functions
that we were unable to \textquotedblleft explain\textquotedblright\ (i.e.,
embed in any infinite family):%
\[
s_{\left(  1,1,1,1\right)  }-s_{\left(  3,1\right)  }+s_{\left(  4\right)
}\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ s_{\left(  4\right)
}-s_{\left(  2,2\right)  }.
\]


While \cref{quest.MNable.which} seems wide open, several particular
cases appear manageable: for example, which products of $h_{i}$'s (or $p_{i}%
$'s) are MNable? Note that the only MNable Schur functions are $h_{i}%
=s_{\left(  i\right)  }$ and $e_{i}=s_{\left(  1,1,\ldots,1\right)  }$.

\acknowledgements{I thank Ira Gessel, Jim Haglund, Linyuan Liu, Patrick Polo, Richard Stanley
and Mark Wildon for interesting and helpful conversations. Parts of this work
were conceived while at the Mathematisches Forschungsinstitut
Oberwolfach as a Leibniz fellow; I thank the MFO for its hospitality. Some
results were discovered using the
SageMath CAS \cite{SageMath}.}

\printbibliography



\end{document}
