%~Mouliné par MaN_auto v.0.29.1 2023-11-02 18:42:34
\documentclass[Unicode, CRMATH, XML]{cedram}

\TopicFR{Combinatoire, Théorie des nombres}
\TopicEN{Combinatorics, Number theory}

\usepackage{amssymb}
\usepackage[noadjust]{cite}

%Changer la largeur des adresses dans les CR (jouer avec \hsize) :
\makeatletter
\def\@setafterauthor{%
  \vglue3mm%
%  \hspace*{0pt}%
\begingroup\hsize=12.5cm\advance\hsize\abstractmarginL\raggedright
\noindent
%\hspace*{\abstractmarginL}\begin{minipage}[t]{10cm}
   \leftskip\abstractmarginL
  \normalfont\Small
  \@afterauthor\par
\endgroup
\vskip2pt plus 3pt minus 1pt
}
\makeatother

\newcommand*\res{\mathcal{R}}

\newcommand*{\tpmod}[1]{\ \ (\mathrm{mod}\ #1)}

\graphicspath{{./figures/}}

\newcommand*{\mk}{\mkern -1mu}
\newcommand*{\Mk}{\mkern -2mu}
\newcommand*{\mK}{\mkern 1mu}
\newcommand*{\MK}{\mkern 2mu}

\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red}

\newcommand*{\relabel}{\renewcommand{\labelenumi}{(\theenumi)}}
\newcommand*{\romanenumi}{\renewcommand*{\theenumi}{\roman{enumi}}\relabel}
\newcommand*{\Romanenumi}{\renewcommand*{\theenumi}{\Roman{enumi}}\relabel}
\newcommand*{\alphenumi}{\renewcommand*{\theenumi}{\alph{enumi}}\relabel}
\newcommand*{\Alphenumi}{\renewcommand*{\theenumi}{\Alph{enumi}}\relabel}
\let\oldtilde\tilde
\renewcommand*{\tilde}[1]{\mathchoice{\widetilde{#1}}{\widetilde{#1}}{\oldtilde{#1}}{\oldtilde{#1}}}
\let\oldexists\exists
\renewcommand*{\exists}{\mathrel{\oldexists}}


\title{Congruences modulo $4$ for the number of $3$-regular partitions}

\author{\firstname{Cristina} \lastname{Ballantine}\IsCorresp}
\address{Department of Mathematics and Computer Science, College of The Holy Cross, Worcester, MA 01610, USA}
\email{cballant@holycross.edu}

\author{\firstname{Mircea} \lastname{Merca}}
\address{Department of Mathematical Methods and Models, Fundamental Sciences Applied in Engineering Research Center, University Politehnica of Bucharest, RO-060042 Bucharest, Romania}
\address{Academy of Romanian Scientists, RO-050044, Bucharest, Romania}
\email{mircea.merca@profinfo.edu.ro}

\begin{abstract}
The last decade has seen an abundance of congruences for $b_\ell(n)$, the number of $\ell$-regular partitions of $n$. Notably absent are congruences modulo $4$ for $b_3(n)$. In this paper, we introduce Ramanujan type congruences modulo $4$ for $b_3(2n)$ involving some primes $p$ congruent to $ 11, 13, 17, 19, 23$ modulo $24$.
\end{abstract}

\keywords{\kwd{partitions}
\kwd{regular partitions}
\kwd{congruences}}
\subjclass{11P83, 05A17, 11F33}


\begin{document}
\maketitle
\section{Introduction}

A partition of a positive integer $n$ is a weakly decreasing sequence of positive integers whose sum is $n$. The positive integers in the sequence are called parts. For more on the theory of partitions, we refer the reader to~\cite{Andrews98}.

For an integer $\ell>1$, a partition is called $\ell$-regular if none of its parts is divisible by $\ell$. The number of the $\ell$-regular partitions of $n$ is usually denoted by $b_\ell(n)$ and its arithmetic properties were investigated extensively. See, for example, \cite{Carlson, Cui, Dand, Furcy, Hirschhorn, Lovejoy, Penn, Penn8, Xia14, Xia15, Wang17, Wang, Webb}. The generating function for $b_\ell(n)$ is given by
\[
\sum_{n=0}^\infty b_\ell(n)\,q^n = \frac{(q^\ell;q^\ell)_\infty}{(q;q)_\infty}.
\]
Here and throughout $q$ is a complex number with $|q| < 1$, and the symbol $(a;q)_\infty$ denotes the infinite product
\[
(a;q)_\infty = \prod_{n=0}^\infty (1-a\,q^n).
\]


In a recent paper, W. J. Keith and F. Zanello~\cite{Keith} discovered infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p$ with $p \equiv 13, 17, 19, 23 \pmod {24}$.

\begin{theo}[Keith--Zanello]\label{Th1}
The sequence $b_3(2n)$ is lacunary modulo $2$. If $p \equiv 13, 17, 19, 23 \pmod {24}$ is prime, then
\[
b_3\big(2(p^2n+pk-24^{-1}) \big)\equiv 0 \pmod 2
\]
for $1 \leqslant k \leqslant p - 1$, where $24^{-1}$ is taken modulo $p^2$.
\end{theo}

Motivated by the Keith--Zanello result, O. X. M. Yao~\cite{Yao} provided new infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p\geqslant 5$.

\begin{theo}[Yao]\label{Th2}
Let $p\geqslant 5$ be a prime.
\begin{enumerate}
\item \label{2.1} If $b_3\left(\frac{p^2-1}{12} \right) \equiv 1 \pmod 2$, then for $n,k\geqslant 0$
\[
b_3\left(2p^{4k+4}\,n+2p^{4k+3}\,j+\frac{p^{4k+4}-1}{12} \right) \equiv 0 \pmod 2
\]
where $1\leqslant j \leqslant p-1$ and for $n,k\geqslant 0$
\[
b_3\left(\frac{p^{4k}-1}{12} \right) \equiv 1 \pmod 2.
\]
\item \label{2.2} If $b_3\left(\frac{p^2-1}{12} \right) \equiv 0 \pmod 2$, then for $n,k\geqslant 0$ with $p \nmid (24n+1)$
\[
b_3\left(2p^{6k+2}\,n+\frac{p^{6k+2}-1}{12} \right) \equiv 0 \pmod 2
\]
and for $n,k\geqslant 0$
\[
b_3\left(\frac{p^{6k}-1}{12} \right) \equiv 1 \pmod 2.
\]
\end{enumerate}
\end{theo}

Very recently, Ballantine, Merca and Radu~\cite{BMR} introduced new infinite Ramanujan type congruences modulo $2$ for $b_3(2n)$.
They complement naturally the results of Keith--Zanello and Yao and involve primes in the set
\[
\mathcal P=\bigl\{p \text{ prime} \,:{} \exists j\in \{1,4,8\},\, x, y \in \mathbb Z,\, \gcd(x,y)=1 \text { with } x^2+216y^2=jp\bigr\}
\]
whose Dirichlet density is $1/6$.

\begin{theo}
For every $p\in \mathcal P$ and $n \geqslant 0$, we have
\begin{equation*}
b_3\big(2\,(p^2\,n+p\,\alpha-24_p^{-1})\big) \equiv 0 \pmod 2,
\end{equation*}
where $0\leqslant \alpha < p$, $\alpha \neq \lfloor p/24 \rfloor$, and $24_p^{-1}$ is the inverse of $24$ modulo $p$ taken such that $1\leqslant -24_p^{-1}\leqslant p-1$.
\end{theo}

In this work, motivated by the results on the parity of $b_3(2n)$, we investigate Ramanujan type congruences modulo $4$ for $b_3(2n)$. We note that congruences modulo $4$ for $\ell$-regular partitions, have been studied in~\cite{Keith15} for $\ell=4, 5, 9$, and in~\cite{R} for $\ell=2$. However, congruences modulo $4$ for $3$-regular partitions are missing from the literature.

\begin{theo}\label{Tmain}
For every $p\in \{43, 47, 59, 61, 67, 89, 137, 139, 157\}$ and $n \geqslant 0$ we have
\begin{equation}\label{eq_main}
b_3\big(2\,(p^2\,n+p\,\alpha-24_p^{-1})\big) \equiv 0 \pmod 4,
\end{equation}
where $0\leqslant \alpha<p$, $\alpha \neq \lfloor p/24\rfloor$, $24_p^{-1}$ is the inverse of $24$ modulo $p$ taken such that $1\leqslant -24_p^{-1}\leqslant p-1$.
\end{theo}

We conjecture that there are infinitely many primes for which~\eqref{eq_main} holds. For example, we verified numerically that, in addition to the primes in Theorem~\ref{Tmain}, the statement of the theorem holds for
\begin{align*}
p\in\{& 167, 181, 229, 233, 277, 331, 359, 379, 401, 419, 421, 431, 443, 479, 499, 541,
\\ & 569, 593, 599, 613, 643, 647, 691, 709, 719, 757, 761, 787, 809, 827, 829, 853,
\\ & 859, 863, 877, 911, 929, 953, 977, 983, 1021, 1031
\}.
\end{align*}
We were unable to finish the proof of Theorem~\ref{Tmain} for these values of $p$ due to computing time limitations.

\section{Proof of Theorem~\ref{Tmain}}

\subsection{Modular forms} As is the case with many proofs of congruences in the literature, we use~\cite[Lemma~4.5]{Radu1a}. For the conveniece of the reader, we first introduce all necessary notation and the statement of~\cite[Lemma~4.5]{Radu1a}. This exposition is nearly identical to that in~\cite{BMR}.

Let $\Gamma:=SL(2, \mathbb Z),$ and define
\[
\Gamma_\infty:=\left\{\left(
\begin{matrix}1& b \\ 0& 1
\end{matrix}\right)\in \Gamma \right\}.
\]
For a positive integer $N$, we define the congruence subgroup
\[
\Gamma_0(N):=\left\{\left(
\begin{matrix}a & b \\ c& d
\end{matrix}\right)\in \Gamma : c\equiv 0 \tpmod N \right\}.
\]
If $M$ is a positive integer, we write $R(M)$ for the set of finite integer sequences $r=(r_{\delta_1}, r_{\delta_2}, \ldots, r_{\delta_k})$, where $1={\delta_1}<{\delta_2}<\cdots <{\delta_k}=M$ are the positive divisors of $M$. We note for the remainder of this section we only consider positive divisors of a given integer.
Given a positive integer $m$, we denote by $S_{24m}$ the set of invertible quadratic residues modulo $24m$ and, for fixed $0\leqslant t\leqslant m-1$, we define
\[
P_{m,r}(t):=\left\{ts+\frac{s-1}{24}\sum_{\delta \mid M}\delta r_{\delta}\tpmod m : s\in S_{24m}\right\}.
\]


Let $m$, $M$ and $N$ be positive integers. Moreover, let $t$ be an integer such that $0\leqslant t\leqslant m-1$ and let $r=(r_\delta)\in R(M)$. We set $\kappa:=\gcd(1-m^2, 24)$ and write $\prod_{\delta\mid M}\delta^{|r_\delta|}=:2^sv$, where $s$ is a nonnegative integer and $v$ is odd. Then, we say that the tuple $(n,M,N,(r_\delta), t)\in \Delta^*$ if and only if all of the following six conditions are satisfied.
\begin{enumerate}
\item \label{c1} $p\mid m$, $p$ prime, implies $p\mid N$;

\item \label{c2} $\delta \mid M$, $\delta \geqslant 1$ such that $r_\delta\neq 0$ implies $\delta \mid mN$;

\item \label{c3} $\kappa N \sum_{\delta \mid M}r_\delta \frac{mN}{\delta} \equiv 0 \pmod{24}$;

\item \label{c4} $\kappa N \sum_{\delta \mid M}r_\delta \equiv 0 \pmod{8}$;\\[-1em]

\item \label{c5} $\frac{24m}{\gcd(\kappa(-24t-\sum_{\delta\mid M}\delta r_\delta),24m)}\Big| N$;\\[-1em]

\item \label{c6} If $2 \mid m$, then $(4\mid \kappa N \text{ and } 8\mid Ns)$ or $(2 \mid s \text{ and } 8 \mid N(1-v))$.
\end{enumerate}
Finally, for $\gamma=\left(
\begin{smallmatrix}a & b \\ c& d
\end{smallmatrix}\right)\in \Gamma$, and $m$ and $r=(r_\delta)\in R(M)$ as above,
we define
\[
p_{m,r}(\gamma):=\min_{d\in\{0, \ldots, m-1\}}\frac{1}{24}\sum_{\delta\mid M}r_\delta\frac{\gcd^2(\delta(a+\kappa dc),mc)}{\delta m}
\]
and for $a=(a_\delta)\in R(N)$, we define
\[
p^*_{a}(\gamma):=\frac{1}{24}\sum_{\delta\mid N}a_\delta\frac{\gcd^2(\delta,c)}{\delta}.
\]

We use the notation
\[
\sum_{n=0}^\infty c(n)\,q^n\equiv \sum_{n=0}^\infty d(n)\,q^n\pmod u
\]
to mean that $c(n)\equiv d(n)\pmod u$ for all $n\geqslant 0$. Similarly,
\[
\sum_{n=0}^\infty c(n)\,q^n\equiv 0 \pmod u
\]
means $c(n)\equiv 0 \pmod u$ for all $n\geqslant 0$.

\begin{lemm}[{\cite[Lemma~4.5]{Radu1a}}] \label{radu}
Let $u$ be a positive integer, $(m,M,N,t,r=(r_\delta))\in \Delta^*$, $a=(a_\delta)\in R(N)$. Let $\{\gamma_1, \ldots, \gamma_n\}\subset \Gamma$ be a complete set of representatives of the double cosets in $\Gamma_0(N)\backslash \Gamma/\Gamma_\infty$. Assume that $p_{m,r}(\gamma_i)+p^*_a(\gamma_i)\geqslant 0$ for all $0\leqslant i\leqslant n$. Let $t_{\min}:=\min_{t'\in P_{m,r}(t)}t'$ and
\[
\nu:=\frac{1}{24}\left(\left(\sum_{\delta \mid N}a_\delta+ \sum_{\delta \mid M}r_\delta\right) [\Gamma:\Gamma_0(N)]-\sum_{\delta \mid N}\delta a_\delta\right)-\frac{1}{24m}\sum_{\delta \mid M}\delta r_\delta-\frac{t_{\min}}{m}.
\]
Suppose
\[
\prod_{\delta\mid M}\prod_{n=1}^\infty (1-q^{\delta n})^{r_\delta}=\sum_{n=0}^\infty c_r(n)q^n.
\]
If
\[
\sum_{n=0}^{\lfloor \nu\rfloor} c_r(mn+t')q^n\equiv 0 \pmod u, \ \ \text{ for all } t'\in P_{m,r}(t),
\]
then
\[
\sum_{n=0}^{\infty} c_r(mn+t')q^n\equiv 0 \pmod u, \ \ \text{ for all } t'\in P_{m,r}(t).
\]
\end{lemm}
\subsection{Proof of Theorem~\ref{Tmain}}
As customary, we use the notation
\[
f_i:=\prod_{k=1}^\infty(1-q^{ki}).
\]
From~\cite{YX13}, identity (2.18), we have
\begin{equation}\label{equivb3}
\sum_{n=0}^\infty b_3(2n)\,q^n=\frac{f_2f_3f_8f_{12}^2}{f_1^2f_4f_6f_{24}}.
\end{equation}
Moreover, since for $i\geq1$, $f_{2i}^2\equiv f_i^4\pmod 4$, we have
\begin{equation}\label{equiv4}
\frac{f_{2i}}{f_{i}}= \frac{f_{2i}^2}{f_{i}^4}\frac{f_{i}^3}{f_{2i}}\equiv \frac{f_{i}^3}{f_{2i}}
\pmod 4.
\end{equation}

Using~\eqref{equivb3} and~\eqref{equiv4}, we obtain
\[
\sum_{n=0}^\infty b_3(2n)\,q^n\equiv
\frac{f_1^2f_3f_4^3f_6^3}{f_2f_8f_{24}}
\pmod 4.
\]



To use the Lemma~\ref{radu}, we write
\[
\sum_{n=0}^\infty c(n)q^n:=\frac{f_1^2f_3f_4^3f_6^3}{f_2f_8f_{24}}=\prod_{\delta\mid M}\prod_{n=1}^\infty (1-q^{\delta n})^{r_\delta}.
\]

Thus, with the notation of Lemma~\ref{radu}, we take $u=4$, $m=p^2$, $M=24$, and
\[
(r_1,r_2,r_3,r_4,r_6,r_8,r_{12},r_{24})=(2,-1,1,3,3,-1,0,-1).
\]
We have $\kappa=24$ and we calculate
\begin{align*}
\sum_{\delta \mid M}\frac{r_\delta}{\delta} = \frac{35}{12},\qquad \sum_{\delta \mid M}{r_\delta} =6,\qquad \sum_{\delta \mid M}{\delta r_\delta} & =1.
\end{align*}
We take $N=24p$. It is straightforward to verify that for any $t=p\alpha-24_p^{-1}$ conditions \eqref{c1}--\eqref{c6} are satisfied.


Since
\[
[\Gamma:\Gamma_0(N)]=N\prod_{x\mid N}(1+x^{-1}),
\]
where the product is taken after all prime divisors of $N$, we have
\[
[\Gamma:\Gamma_0(N)]=48(p+1).
\]

In general, it is nontrivial to find a complete set of representatives for the double cosets in $\Gamma_0(N)\setminus \Gamma/\Gamma_\infty$. If $N$ is square free, it is shown in~\cite[Lemma~2.6]{RS1} that a complete set of representatives for $\Gamma_0(N)\backslash \Gamma/\Gamma_\infty$ is given by
\[
\mathcal A_N=\left\{\left(
\begin{matrix}1 & 0 \\ \delta & 1
\end{matrix}\right): \delta \mid N, \, \delta \geqslant 1\right\}.
\]
This result has been extended to $N$ such that $N/2$ is square free in~\cite[Lemma~4.3]{Wang17a}. While for $N=24p$, neither $N$ nor $N/2$ is square free, when $m=p^2, \kappa=24$, and $(r_1,r_2,r_3,r_4,r_6,r_8,r_{12},r_{24})=(2,-1,1,3,3,-1,0,-1)$ we can avoid finding a complete set of representatives all together.


Since for any integers $i,j\geq 1$ we have
\[
\gcd(j(a+\kappa dc),mc)\leqslant\gcd(ij(a+\kappa dc),mc)\leqslant i\gcd(j(a+\kappa dc),mc),
\]
an easy calculation shows that for each $\gamma= \left(
\begin{smallmatrix}a & b \\ c & d
\end{smallmatrix}\right)\in \Gamma$, we have
\begin{align*}
\sum_{\delta\mid M}r_\delta\frac{\gcd^2(\delta(a+\kappa dc),mc)}{\delta m}\geq0,
\end{align*}
and thus $p_{m,r}(\gamma)\geq 0$. Hence, we can use $a: = (a_\delta)_{\delta \mid N}$ with $a_\delta=0$ for each $\delta \mid N$ to calculate $\lfloor \nu \rfloor$. It is clear from the definition of $\nu$ in Lemma~\ref{radu} that
\[
\lfloor \nu \rfloor= 12\,(p+1)-1.
\]

Let
\[
\res_p:=\{p\alpha-24_p^{-1} : 0\leqslant \alpha <p, \ \alpha\neq \lfloor p/24 \rfloor\}.
\]
For each $p$, we used Mathematica\textsuperscript{\textsc{tm}} to write the set $\res_p$ as
\[
\res_p=P_{m,r}(-24_p^{-1})\cup P_{m,r}(Ap-24_p^{-1})
\]
for a minimal $A$. For example, when $p=43$, we have $A=2$ and the Mathematica\textsuperscript{\textsc{tm}} calculation gives
\begin{align*}
P_{m,r}(-24_p^{-1})=\{& 34, 163, 206, 292, 378, 421, 593, 851, 894, 937, 1023, 1195,\\ & 1238,
1281, 1324, 1367, 1453, 1496, 1539, 1668, 1754 \}
\end{align*}
and
\begin{align*}
P_{m,r}(2\cdot 43-24_p^{-1})=\{& 120, 249, 335, 464, 507, 550, 636, 679, 722, 765, 808, 980, \\ & 1066,
1109, 1152, 1410, 1582, 1625, 1711, 1797, 1840\}.
\end{align*}
If $\alpha = \lfloor 43/24\rfloor=1$ we have $P_{m,r}(p-24_p^{-1})=P_{m,r}(77)=\{77\}.$



For $p\in\{43, 47, 139, 157\}$ we obtained $A=2$ and for $p\in\{59, 61, 67, 89, 137\}$ we obtained $A=1$. Moreover, if $t^*=p\lfloor p/24\rfloor-24^{-1}_p$, then $P_{m,r}(t^*)=\{t^*\}$.

Finally, in each case, we verified that for each $t\in \res_p$ we have
\[
c(p^2n+t)\equiv 0\pmod 4 \ \ \text{ for } 0\leqslant n\leqslant 12(p+1)-1.
\]

Then, \cite[Lemma~4.5]{Radu1a} implies that, for each
\[
p\in \mathcal U:=\{43, 47, 59, 61, 67, 89, 137, 139, 157\}
\]
we have
\[
c(p^2n+t)\equiv 0\pmod 4 \ \ \text{ for all } n\geqslant 0 \text{ and } t\in \res_p.
\]

Our calculations show that for each prime $p\in \mathcal U$, we have that
\[
c(p\lfloor p/24\rfloor-24^{-1}_p)\not \equiv 0 \pmod 4
\]
and thus the requirement that $\alpha\neq \lfloor p/24\rfloor$ in the statement of Theorem~\ref{Tmain} is necessary.

\bigskip
\section{Concluding remarks}

Several Ramanujan type congruences modulo $4$ for $b_3(2n)$ involving some primes $p$ with $p\equiv 11, 13, 17, 19, 23 \pmod {24}$ have been proved in this paper using modular forms and~\cite[Lemma~4.5]{Radu1a}.

As mentioned in the introduction, we verified the statement of Theorem~\ref{Tmain} numerically up to $10^8$ for many more primes equivalent to $ 11, 13, 17, 19, 23$ modulo $24$. In fact, our computations suggest that~\eqref{eq_main} is also true for primes $p=1033$ and $p=1153$ which are congruent to $1$ modulo $24$. We did not encounter any primes congruent to $5$ or $7$ modulo $24$ for which~\eqref{eq_main} holds. We leave it as an open problem to characterize an infinite family of primes for which the statement of Theorem~\ref{Tmain} holds and to prove the theorem for all these primes.

\bigskip
\bibliographystyle{crplain}
\bibliography{CRMATH_Ballantine_20230273}
\end{document}