1 Introduction and statement of results
A partition of a positive integer n is any nonincreasing sequence of positive integers, called parts, whose sum is n. Let
$\ell $
be a fixed positive integer. An
$\ell $
-regular partition of a positive integer n is a partition of n such that none of its parts is divisible by
$\ell $
. Let
$b_{\ell }(n)$
be the number of
$\ell $
-regular partitions of n. The generating function for
$b_{\ell }(n)$
is
$$ \begin{align*} \sum_{n=0}^{\infty}b_{\ell}(n)q^n=\frac{f_{\ell}}{f_1}, \end{align*} $$
where
$f_k:=(q^k; q^k)_{\infty }=\prod _{j=1}^{\infty }(1-q^{\:jk})$
and k is a positive integer.
In 2018, Hirschhorn [Reference Gireesh and Mahadeva Naika3] derived a number of congruences for
$p_3(n)$
modulo high powers of 3, where
$p_3(n)$
denotes the number of partitions of n in three colours. Let
$p_{\{3, 3\}}(n)$
denote the number of
$3$
-regular partitions in three colours, whose generating function is
$$ \begin{align} \sum_{n=0}^{\infty}p_{\{3,3\}}(n)q^n=\frac{f_3^3}{f_1^3}. \end{align} $$
In 2019, Gireesh and Mahadeva Naika [Reference Hirschhorn2] studied the function
$p_{\{3, 3\}}(n)$
and deduced some congruences modulo powers of 3 for
$p_{\{3, 3\}}(n)$
. In a very recent paper [Reference da Silva and Sellers1], using elementary generating function manipulations and classical techniques, da Silva and Sellers significantly extended the list of proven arithmetic properties satisfied by
$p_{\{3, 3\}}(n)$
. They gave a parity characterisation for
$p_{\{3, 3\}}(2n)$
and a complete characterisation for
$p_{\{3, 3\}}(n)$
modulo 3. For example, they proved that, for all
$n\geq 0$
,
$$ \begin{align*} p_{\{3, 3\}}(3n+1)&\equiv p_{\{3, 3\}}(3n+2)\equiv 0\ \pmod 3,\\ p_{\{3, 3\}}(3n)&\equiv \begin{cases} (-1)^{k+\ell}\ \pmod{3} & \text{if}\ n=k(3k-1)/2+\ell(3\ell -1)/2,\\ 0\ \pmod{3} & \text{otherwise}.\nonumber \end{cases} \end{align*} $$
They also found some congruences for
$p_{\{3, 3\}}(n)$
modulo
$4$
and
$9$
. They further conjectured four Ramanujan-like congruences modulo 5 satisfied by
$p_{\{3, 3\}}(n)$
.
Conjecture 1.1 [Reference da Silva and Sellers1, Conjecture 5.1]
For all
$n\geq 0$
,
$$ \begin{align} p_{\{3, 3\}}(15n+6)&\equiv 0\ \pmod{5}, \end{align} $$
$$ \begin{align} p_{\{3, 3\}}(25n+6)&\equiv 0\ \pmod{5}, \end{align} $$
$$ \begin{align} p_{\{3, 3\}}(25n+16)&\equiv 0\ \pmod{5}, {\kern3pt} \end{align} $$
$$ \begin{align} p_{\{3, 3\}}(25n+21)&\equiv 0\ \pmod{5}. {\kern6pt} \end{align} $$
Using the theory of modular forms, we confirm that Conjecture 1.1 is true.
Theorem 1.2. Conjecture 1.1 is true.
2 Proof of Theorem 1.2
We prove Theorem 1.2 using the approach developed in [Reference Radu4, Reference Radu and Sellers5]. We define the matrix groups
$$ \begin{align*} \Gamma :=\left\{\begin{bmatrix} a & b \\ c & d \end{bmatrix}: a, b, c, d \in \mathbb{Z}, ad-bc=1 \right\}, \quad \Gamma_{\infty} :=\left\{ \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} \in \Gamma : n\in \mathbb{Z} \right\}. \end{align*} $$
For a positive integer N, let
$$ \begin{align*}\Gamma_{0}(N) :=\left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in \Gamma : c\equiv~0\ \pmod N \right\}.\end{align*} $$
The index of
$\Gamma _{0}(N)$
in
$\Gamma $
is
$$ \begin{align*} [\Gamma : \Gamma_0(N)] = N\prod_{p|N}(1+p^{-1}), \end{align*} $$
where p is a prime divisor of N.
We now recall some of the definitions and results from [Reference Radu4, Reference Radu and Sellers5] that will be used to prove our results. For a positive integer M, let
$R(M)$
be the set of integer sequences
$r=(r_\delta )_{\delta |M}$
indexed by the positive divisors of M. If
$r \in R(M)$
and
$1=\delta _1<\delta _2< \cdots <\delta _k=M$
are the positive divisors of M, we write
$r=(r_{\delta _1}, \ldots , r_{\delta _k})$
. Define
$c_r(n)$
by
$$ \begin{align*} \sum_{n=0}^{\infty}c_r(n)q^n:=\prod_{\delta|M}(q^{\delta};q^{\delta})^{r_{\delta}}_{\infty}=\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{n \delta})^{r_{\delta}}. \end{align*} $$
The approach to proving congruences for
$c_r(n)$
developed by Radu [Reference Radu4, Reference Radu and Sellers5] reduces the number of coefficients that one must check as compared with the classical method that uses Sturm’s bound alone.
Let m be a positive integer. For any integer s, let
$[s]_m$
denote the residue class of s in
$\mathbb {Z}_m:= \mathbb {Z}/ {m\mathbb {Z}}$
. Let
$\mathbb {Z}_m^{*}$
be the set of all invertible elements in
$\mathbb {Z}_m$
. Let
$\mathbb {S}_m\subseteq \mathbb {Z}_m$
be the set of all squares in
$\mathbb {Z}_m^{*}$
. For
$t\in \{0, 1, \ldots , m-1\}$
and
$r \in R(M)$
, we define a subset
$P_{m,r}(t)\subseteq \{0, 1, \ldots , m-1\}$
by
$$ \begin{align*} P_{m,r}(t):=\bigg\{t': \text{ there exists } [s]_{24m}\in \mathbb{S}_{24m} ~ \text{such} ~ \text{that} ~ t'\equiv ts+\frac{s-1}{24}\sum_{\delta|M}\delta r_\delta \ \pmod{m} \bigg\}. \end{align*} $$
Definition 2.1. Suppose that
$m, M$
and N are positive integers,
$r=(r_{\delta })\in R(M)$
and
$t\in \{0, 1, \ldots , m-1\}$
. Let
$k=k(m):=\gcd (m^2-1,24)$
and write
$$ \begin{align*} \prod_{\delta|M}\delta^{|r_{\delta}|}=2^s\cdot j, \end{align*} $$
where s and j are nonnegative integers with j odd. The set
$\Delta ^{*}$
consists of all tuples
$(m, M, N, (r_{\delta }), t)$
satisfying these conditions and all of the following:
-
(1) each prime divisor of m is also a divisor of N;
-
(2)
$\delta \mid M$
implies that
$\delta \mid mN$
for every
$\delta \geq 1$
such that
$r_{\delta } \neq 0$
; -
(3)
$kN\sum _{\delta |M}r_{\delta } mN/\delta \equiv 0 \pmod {24}$
; -
(4)
$kN\sum _{\delta |M}r_{\delta } \equiv 0 \pmod {8}$
; -
(5)
${24m}/{\gcd {(-24kt-k{\sum _{{\delta }|M}}{\delta r_{\delta }}},24m)}$
divides N; -
(6) if
$2 \mid m$
, then either
$4 \mid kN$
and
$8 \mid sN$
or
$2 \mid s$
and
$8 \mid (1-j)N$
.
Let
$m, M$
and N be positive integers. For
$\gamma = [\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}] \in \Gamma $
,
$r\in R(M)$
and
$r'\in R(N)$
, set
$$ \begin{align*} p_{m,r}(\gamma):=\min_{\lambda\in\{0, 1, \ldots, m-1\}}\frac{1}{24}\sum_{\delta|M}r_{\delta}\frac{\gcd^2(\delta a+ \delta k\lambda c, mc)}{\delta m} \end{align*} $$
and
$$ \begin{align*} p_{r'}^{*}(\gamma):=\frac{1}{24}\sum_{\delta|N}r^{\prime}_{\delta}\frac{\gcd^2(\delta, c)}{\delta}. \end{align*} $$
Lemma 2.2 [Reference Radu4, Lemma 4.5]
Let u be a positive integer,
$(m, M, N, r=(r_{\delta }), t)\in \Delta ^{*}$
and
$r'=(r^{\prime }_{\delta })\in R(N)$
. Let
$\{\gamma _1,\gamma _2, \ldots , \gamma _n\}\subseteq \Gamma $
be a complete set of representatives of the double cosets of
$\Gamma _{0}(N) \backslash \Gamma / \Gamma _\infty $
. Assume that
$p_{m,r}(\gamma _i)+p_{r'}^{*}(\gamma _i) \geq 0$
for all
$1 \leq i \leq n$
. Let
$t_{\mathrm{min}}=\mathrm{min}_{t' \in P_{m,r}(t)} t'$
and
$$ \begin{align*} \nu:= \frac{1}{24}\bigg\{ \bigg( \sum_{\delta|M}r_{\delta}+\sum_{\delta|N}r^{\prime}_{\delta}\bigg)[\Gamma:\Gamma_{0}(N)] -\sum_{\delta|N} \delta r^{\prime}_{\delta}\bigg\}-\frac{1}{24m}\sum_{\delta|M}\delta r_{\delta} - \frac{ t_{\mathrm{min}}}{m}. \end{align*} $$
If the congruence
$c_r(mn+t')\equiv 0 \pmod u$
holds for all
$t' \in P_{m,r}(t)$
and
$0\leq n\leq \lfloor \nu \rfloor $
, then it holds for all
$t'\in P_{m,r}(t)$
and
$n\geq 0$
.
To apply the above lemma, we need the following result that gives us a complete set of representatives of the double cosets in
$\Gamma _{0}(N) \backslash \Gamma / \Gamma _\infty $
.
Lemma 2.3 [Reference Wang6, Lemma 4.3]
If N or
$\tfrac 12N$
is a square-free integer, then
$$ \begin{align*} \bigcup_{\delta|N}\Gamma_0(N)\begin{bmatrix} 1 & 0 \\ \delta & 1 \end{bmatrix}\Gamma_ {\infty}=\Gamma. \end{align*} $$
Proof of Theorem 1.2
From (1.1),
$$ \begin{align*} \sum_{n=0}^{\infty}p_{\{3,3\}}(n)q^n=\frac{(q^3; q^3)_{\infty}^3}{(q; q)_{\infty}^3}. \end{align*} $$
We choose
$(m,M,N,r,t)=(15,3,15,(-3,3),6)$
. We verify that
$(m,M,N,r,t) \in \Delta ^{*}$
and
$P_{m,r}(t)=\{6\}$
. By Lemma 2.3,
$\{[\begin {smallmatrix} 1 & 0 \\ \delta & 1 \end {smallmatrix}]:\delta \mid 15 \}$
forms a complete set of double coset representatives of
$\Gamma _{0}(N) \backslash \Gamma / \Gamma _\infty $
. Let
$\gamma _{\delta }=[\begin {smallmatrix} 1 & 0 \\ \delta & 1 \end {smallmatrix}]$
and
$r'=(30,0,0,0)\in R(15)$
. We use Sage to verify that
$p_{m,r}(\gamma _{\delta })+p_{r'}^{*}(\gamma _{\delta }) \geq 0$
for each
$\delta \mid N$
. From Lemma 2.2,
$$ \begin{align*} \nu&:= \frac{1}{24}\bigg\{ \bigg( \sum_{\delta|M}r_{\delta}+\sum_{\delta|N}r^{\prime}_{\delta}\bigg)[\Gamma:\Gamma_{0}(N)] -\sum_{\delta|N} \delta r^{\prime}_{\delta}\bigg\}-\frac{1}{24m}\sum_{\delta|M}\delta r_{\delta} - \frac{ t_{\mathrm{min}}}{m}\\ &\;=\frac{1}{24}\{ ( 0+30)24 -30\}-\frac{1}{24\cdot 15}(-3+9) - \frac{ 6}{15}=\frac{85}{3}. \end{align*} $$
Therefore,
$\lfloor \nu \rfloor $
is equal to
$28$
. Using Sage, we verify that
$p_{\{3,3\}}(15n+6)\equiv 0 \pmod {5}$
for
$0\leq n\leq 28$
. By Lemma 2.2, we conclude that
$p_{\{3,3\}}(15n+6)\equiv 0 \pmod {5}$
for all
$n\geq 0$
. This completes the proof of (1.2).
To prove (1.3), we take
$(m,M,N,r,t)=(25,3,15,(-3,3),6)$
. It is easy to verify that
$(m,M,N,r,t) \in \Delta ^{*}$
and
$P_{m,r}(t)=\{6\}$
. We compute the upper bound in Lemma 2.2 to be
$\lfloor \nu \rfloor =47$
. In the same way as for (1.2), we find that
$p_{\{3,3\}}(25n+6)\equiv 0 \pmod {5}$
for all
$n\geq 0$
.
We now prove (1.4) and (1.5). We take
$(m,M,N,r,t)=(25,3,15,(-3,3),16)$
. It is easy to verify that
$(m,M,N,r,t) \in \Delta ^{*}$
and
$P_{m,r}(t)=\{16,21\}$
. Here we also check that
$t_{\textrm {min}}=16$
. By Lemma 2.3,
$\{[\begin {smallmatrix} 1 & 0 \\ \delta & 1 \end {smallmatrix}]:\delta \mid 15 \}$
forms a complete set of double coset representatives of
$\Gamma _{0}(N) \backslash \Gamma / \Gamma _\infty $
. Let
$\gamma _{\delta }=[\begin {smallmatrix} 1 & 0 \\ \delta & 1 \end {smallmatrix}]$
and
$r'=(50,0,0,0)\in R(15)$
. We use Sage to verify that
$p_{m,r}(\gamma _{\delta })+p_{r'}^{*}(\gamma _{\delta }) \geq 0$
for each
$\delta \mid N$
. We compute the upper bound in Lemma 2.2 to be
$\lfloor \nu \rfloor =47$
. Using Sage, we verify that
$p_{\{3,3\}}(25n+t')\equiv 0 \pmod {5}$
for all
$t' \in P_{m,r}(t)$
and for
$0\leq n\leq 47$
. By Lemma 2.2, we conclude that
$p_{\{3,3\}}(25n+t')\equiv 0 \pmod {5}$
for all
$t' \in P_{m,r}(t)$
and for all
$n\geq 0$
. This completes the proof of the theorem.
