Divisibility properties of weighted $k$ regular partitions

TL;DR

This paper explores a generalized class of weighted k-regular partitions, establishing new Ramanujan-type congruences, divisibility results, and prime sets where the partition function vanishes modulo primes.

Contribution

It introduces a broad generalization of k-regular partitions and proves novel congruences and divisibility properties extending classical results.

Findings

Established new Ramanujan-type congruences

Identified divisibility results for the partition functions

Found prime sets with vanishing partition values modulo primes

Abstract

We study a generalized class of weighted $k$-regular partitions defined by \[ \sum_{n=0}^{\infty} c_{k, r_1, r_2}(n) q^n = \prod_{n=1}^{\infty} \frac{(1 - q^{nk})^{r_1}}{(1 - q^n)^{r_2}}, \] which extends the classical $k$-regular partition function $b_k(n)$. We establish new infinite families of Ramanujan-type congruences, divisibility results, and positive-density prime sets for which $c_{k, r_1, r_2}(n)$ vanishes modulo a given prime.

Equations

$$\sum _{n=0}^{\infty }{c}_{k,r_1,r_2}(n)q^n=\prod _{n=1}^{\infty }\frac{(1-q^{nk}{)}^{r_1}}{(1-q^n{)}^{r_2}},$$

$$\sum _{n=0}^{\infty }{b}_k(n)q^n=\prod _{n=1}^{\infty }\frac{1-q^{nk}}{1-q^n}.$$

$$\sum _{n=0}^{\infty }{c}_{k,r_1,r_2}(n)q^n=\prod _{n=1}^{\infty }\frac{(1-q^{nk}{)}^{r_1}}{(1-q^n{)}^{r_2}}.$$

$$c_{p,r,Mr}\left(\frac{dmn\ell -r(p-M)}{24}\right)\equiv 0(\mathrm{mod}m),$$

$$b_p\left(\frac{dmn\ell -(p-1)}{24}\right)\equiv 0(\mathrm{mod}m),$$

$$\chi (d)f(\tau )=f{|}_{\kappa }\gamma (z):=(c\tau +d{)}^{-\kappa }f\left(\frac{a\tau +b}{c\tau +d}\right),$$

$$f{|}_{\kappa }\gamma (\tau )=\sum _{n\gg -\infty }{a}_{f,\alpha }(n)q^{\frac{n}{M}}$$

$$\eta (\tau ):=q^{1/24}\prod _{n=1}^{\infty }(1-q^n),\text{where }q=e^{2\pi i\tau },\Im (z)>0.$$

$$f(\tau )=\prod _{\delta \mid N}{\eta }^{r_{\delta }}(\delta \tau ),$$

$$\sum _{\delta \mid N}\delta r_{\delta }\equiv 0(\mathrm{mod}24),\text{and}\sum _{\delta \mid N}\frac{N}{\delta }{r}_{\delta }\equiv 0(\mathrm{mod}24).$$

$$f(A\tau )=\chi (d)(c\tau +d{)}^{\kappa }f(\tau ),$$

$$\kappa =\frac{1}{2}\sum _{\delta \mid N}{r}_{\delta },$$

$$\chi (d)=\left(\frac{(-1{)}^{\kappa }s}{d}\right),\text{with}s=\prod _{\delta \mid N}{\delta }^{r_{\delta }}.$$

$$\frac{1}{24}\cdot \frac{N}{d\cdot \gcd (d,N/d)}\sum _{\delta \mid N}\frac{r_{\delta }}{\delta }\cdot \gcd (d,\delta {)}^2.$$

$$f\mid T(\ell )\equiv 0(\mathrm{mod}m),$$

$$f(t\tau )=\sum _{n=0}^{\infty }a(n)q^{tn},$$

$$f(\tau )|T(p):=\sum _{n=0}^{\infty }(a(pn)+\psi (p)p^{k-1}a\left(\frac{n}{p}\right))q^n,$$

$$f(\tau )|T(p)\equiv f(\tau )|U(p)(\mathrm{mod}p),$$

$$\left(\sum _{n=0}^{\infty }a(n)q^n\right)|U(p):=\sum _{n\equiv 0(\mathrm{mod}p)}a(n)q^{n/p}.$$

$$\eta \left(-\frac{1}{\tau }\right)=(-i\tau {)}^{1/2}\eta (\tau ),$$

$$q=\exp \left(\frac{2\pi i}{k}(h+iw)\right),$$

$$q_1=\exp \left(\frac{2\pi i}{k}\left(h^{\prime }+i{w}^{-1}\right)\right).$$

$$\eta \left(\frac{1}{k}(h+iw)\right)$$

$$\eta \left(\frac{1}{k}(h^{\prime }+i{w}^{-1})\right)$$

$$w_{h,k}=\left(\frac{-h}{k}\right)\exp \left(-\pi i\left[\frac{1}{4}(k-1)+\frac{1}{12}\left(k-\frac{1}{k}\right)(h-h^{\prime })\right]\right).$$

$$f_{p,m,r_1,r_2}(\tau ):=\left(\frac{\eta (p\tau {)}^{r_1}}{\eta (\tau {)}^{r_2}}\right)\eta (pm\tau {)}^a\eta (m\tau {)}^b,$$

$$(1-x^p{)}^m\equiv (1-x^{pm})(\mathrm{mod}m),$$

$$f_{p,m,r_1,r_2}(\tau )\equiv \eta (p\tau {)}^{am+r_1}\eta (\tau {)}^{bm-r_2}(\mathrm{mod}m).$$

$$m(pa+b)\equiv r_2-p{r}_1(\mathrm{mod}24),$$

$$m(a+pb)\equiv p{r}_2-r_1(\mathrm{mod}24).$$

$${\eta }^{am+r}(p\tau ){\eta }^{bm-Mr}(\tau )\in S_{\kappa }({\Gamma }_0(p),{\chi }_p),$$

Full text

[TABLE]

which extends the classical $k$-regular partition function $b_{k}(n)$. We establish new infinite families of Ramanujan-type congruences, divisibility results, and positive-density prime sets for which $c_{k,r_{1},r_{2}}(n)$ vanishes modulo a given prime.

2010 Mathematics Subject Classification. Primary 05A17, 11P81, 11P83, Secondary 11F11.

Keywords and phrases. Modular form, cusp form, $k$-Regular partitions, congruence.

1. Introduction

Partition theory plays a central role in number theory and combinatorics, with origins in the classical work of Euler, Ramanujan, and Hardy. Partitions of $n$ are non-increasing sequences of positive integers (parts) which sum to $n$. Among the various classes of partitions, the family of $k$-regular partitions—those in which no part is divisible by a fixed integer $k\geq 1$—has been extensively studied. The generating function for the number of such partitions of an integer $n$, denoted by $b_{k}(n)$, is given by

[TABLE]

This family generalizes the classical partition function $p(n)$, which counts the number of partitions of $n$, since $b_{k}(n)=p(n)$ for $k>n$. and arises naturally in the theory of modular forms, $q$-series, and congruences in arithmetic functions. Moreover, when $k=p$ is prime, $b_{p}(n)$ coincides with the number of irreducible $p$-modular representations of the symmetric group $S_{n}$ [6].

Pioneering results on the congruence properties of $k$-regular partitions were established by Gordon and Hughes [2], and further extended in later works by Gordon and Ono [3], Hirschhorn and Sellers [5], and Lovejoy [7]. These investigations, together with Martin’s theory of multiplicative $\eta$-quotients [9], have revealed deep connections between partition functions and modular forms. The modular approach has been developed extensively in the works of Murty [10], and Ono [14], with modern surveys such as Ono and Webb [15] offering a comprehensive perspective. In particular, Mahlburg [8] extended Ramanujan-type congruences using $p$-adic modular forms, opening the way for further density and divisibility results.

The arithmetic and asymptotic behavior of $k$-regular partitions have also been studied from analytic and Galois-theoretic perspectives. Serre [16] showed that many partition functions modulo primes are modular and exhibit congruences on positive density sets. Ahlgren and Ono [1] investigated congruences and density results for partition functions using cusp forms and Galois representations. More recently, Zheng [19] examined the divisibility and distribution of the $3$-regular and $5$-regular partition functions $b_{3}(n)$ and $b_{5}(n)$, investigating both regular and irregular behaviors in its residue classes modulo primes.

Beyond the standard $k$-regular partitions, one can consider generalized or weighted partition functions that extend the form of (1.1). Let $r_{1}\geq 1$ and $r_{2}\geq 1$ be integers. Define the function $c_{k,r_{1},r_{2}}(n)$ via the generating function

[TABLE]

If $r_{2}\geq r_{1}$, then $c_{k,r_{1},r_{2}}(n)$ counts the number of weighted partitions of $n$ in which parts divisible by $k$ appear in $r_{2}-r_{1}$ colors, and all other parts appear in $r_{2}$ colors. Clearly, for $r_{1}=r_{2}=1$, we recover $b_{k}(n)$.

These generalized partition functions arise in the study of combinatorial generating functions, identities of the Rogers–Ramanujan type, and in asymptotic enumeration problems. Although we assume that $r_{2}\geq r_{1}$ throughout this paper, we note that $c_{k,k,1}(n)$ counts the number of $k$-core partitions of $n$, which were shown by Ono [12, 13] and Granville–Ono [4] to be positive for $k\geq 4$ (i.e., $c_{k,k,1}(n)>0$ for $k\geq 4$ and $n\in\mathbb{N}$). Taking $r_{1}=r_{2}=r$, one recovers the $k$-th infinite Borwein product $\prod_{n\geq 1}\frac{1-q^{n}}{1-q^{kn}}$ taken to the power $-r$; if one allows negative $r$, then signs of these Fourier coefficients were considered as special cases of results of Schlosser and Zhou [17].

In this paper, we extend the study of $5$-regular partitions by analyzing the arithmetic and asymptotic behavior of the generalized partition function $c_{k,r_{1},r_{2}}(n)$. Our aim to is build upon the results of Zheng  [19] in order to obtain congruences satisfied by the weighted $k$-regular partition function $c_{k,r_{1},r_{2}}(n)$ modulo primes.

Theorem 1.1**.**

Let $p$ be a prime and $M\geq 1$ an odd integer satisfying $p\geq M$. Let $m\geq 5$ be a sufficiently large prime, such that $\frac{(p-1)(m-1)}{4}$ is even and let $r\geq 1$ be an integer. Let $c_{p,r,Mr}(n)$ be as defined in (1.2), and set $s=\gcd(r,24)$ and $d=\gcd\!\left(\frac{24}{s},\,p-M\right)$. Then there exists a set of primes $\ell$ of positive density such that

[TABLE]

for all integers $n$ with $\gcd(n,\ell)=1$.

In particular, if we put $r=M=1$, we obtain the following corollary.

Corollary 1.2**.**

Let $p$ be a prime and let $m\geq 5$ be a sufficiently large prime. Let $b_{k}(n)$ be as defined in (1.1), and set $d=\gcd(24,\,p-1)$. Then there exists a set of primes $\ell$ of positive density such that

[TABLE]

for all integers $n$ with $\gcd(n,\ell)=1$.

In particular, when $p\in\{3,5\}$, we recover [19, Theorems 2 and 3]. The proof mostly follows the argument of Zheng [19], with one major technical difficulty. In Zheng’s case, the minimal order of vanishing coming from [19, Theorem 5] immediately implies that a certain ratio is a cusp form because of the level. In our more general setting, this does not immediately follow because the level goes to infinity. As a result, we need to more carefully analyze the order of vanishing at all cusps of the image of a cusp form under the Hecke operator, showing that the main term vanishes (see the calculations leading up to (4.40)). The other steps follow from Zheng’s argument in [19] and known results in the theory of modular forms.

The paper is organized as follows. In Section 2, preliminaries about eta-products and modular forms are recalled. In Section 3, we construct certain cusp forms which play an important role in our proofs. Finally, we compute the order of vanishing at all cusps under the Hecke operator and prove Theorem 1.1 in Section 4.

Acknowledgements

The research of the first author was funded by the SERB under File No. MTR/2023/000837 and CRG/2023/002698. The research of the second author was supported by grants from the Research Grants Council of the Hong Kong SAR, China (project numbers HKU 17314122, HKU 17305923).

2. Preliminaries

In this section, we collect foundational results and notation related to eta-quotients and $k$-regular partitions that will be used throughout the paper.

2.1. Modular forms

Recall that, for $\kappa,N\in\mathbb{N}$ and a character $\chi$ modulo $N$, a holomorphic function $f$ from the complex upper half-plane $\mathbb{H}$ to $\mathbb{C}$ is called a modular form of weight $\kappa$ and Nebentypus character $\chi$ if for every $\gamma=\left[\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right]\in\Gamma_{0}(N)$ it satisfies

[TABLE]

and $f|_{\kappa}\gamma(\tau)$ grows at most polynomially as $\tau\to i\infty$ for every $\gamma\in\operatorname{SL}_{2}(\mathbb{Z})$. More generally, for $\kappa\in\mathbb{Z}$ we say that a function $f:\mathbb{H}\to\mathbb{C}$ satisfies weight $k$ modularity on $\Gamma_{0}(N)$ with character $\chi$ if (2.1) holds. We call the equivalence classes in $\Gamma\backslash(\mathbb{Q}\cup\{i\infty\})$ the cusps of $\Gamma$, and sometimes abuse notation to call elements of $\mathbb{Q}\cup\{i\infty\}$ the cusps of $\Gamma$. The condition that $f|_{\kappa}\gamma(z)$ grows at most polynomially as $\tau\to i\infty$ may then be considered a growth condition of $f$ towards the cusp $\gamma(i\infty)=\frac{a}{c}$. For a cusp $\alpha=\gamma(i\infty)$ and $q:=e^{2\pi i\tau}$, we call

[TABLE]

the Fourier expansion of $f$ at $\alpha$. Here $M$ is called the cusp width of $\alpha$ and may be chosen minimally so that $T^{M}\in\gamma^{-1}\Gamma\gamma$. We omit $\alpha$ in the notation when it is $i\infty$ and sometimes omit $f$ when it is clear from context.

2.2. Eta-Quotients and Modularity

Let $\eta(z)$ denote the Dedekind eta-function, defined by

[TABLE]

An eta-quotient is a function of the form

[TABLE]

for integers $r_{\delta}\in\mathbb{Z}$ indexed by the positive divisors $\delta$ of some fixed $N\in\mathbb{N}$. The modular properties of such functions are described by the following result.

Theorem 2.1** (Gordon–Hughes–Newman).**

Let $f(\tau)=\prod_{\delta\mid N}\eta^{r_{\delta}}(\delta\tau)$ be an eta-quotient satisfying the conditions

[TABLE]

Then $f(\tau)$ transforms under $\Gamma_{0}(N)$ as

[TABLE]

for all $A=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\in\Gamma_{0}(N)$, where the weight is

[TABLE]

and the character $\chi$ is given by

[TABLE]

In other words, $f$ satisfies weight $\kappa$ modularity on $\Gamma_{0}(N)$ with character $\chi$.

The behavior of eta-quotients at the cusps can be described explicitly using the following result.

Theorem 2.2** (Ligozat).**

Let $f(\tau)=\prod_{\delta\mid N}\eta^{r_{\delta}}(\delta\tau)$ be an eta-quotient satisfying the modularity conditions (2.5). Let $d\mid N$, and let $c$ be a positive integer with $\gcd(c,d)=1$. Then the order of vanishing of $f(\tau)$ at the cusp $c/d$ is

[TABLE]

We need a famous theorem of Serre about congruences for the Hecke operators.

Theorem 2.3** (J.-P. Serre).**

The set of primes $\ell\equiv-1\pmod{Nm}$ such that

[TABLE]

for each $f(\tau)\in S_{k}(\Gamma_{0}(N),\psi)_{m}$ has positive density, where $T(\ell)$ denotes the usual Hecke operator acting on $S_{k}(\Gamma_{0}(N),\psi)$.

Before proving the key result, we record some useful facts about modular forms and their Fourier coefficients. For a more detailed overview, the reader is referred to [11].

Proposition 2.4**.**

Let $f(\tau)=\sum_{n=0}^{\infty}a(n)q^{n}$ be a modular form in $M_{k}(\Gamma_{0}(N),\psi)$.

1. (1)

For any positive integer $t$, the function

[TABLE]

is the Fourier expansion of a modular form in $M_{k}(\Gamma_{0}(tN),\psi)$. 2. (2)

For any prime $p$, the function

[TABLE]

is the Fourier expansion of a modular form in $M_{k}(\Gamma_{0}(N),\psi)$.

Moreover, both assertions remain valid when $M_{k}(\Gamma_{0}(N),\psi)$ is replaced by $S_{k}(\Gamma_{0}(N),\psi)$.

Here $T(p)$ denotes the Hecke operator associated with the prime $p$. In particular, one has the congruence

[TABLE]

in $M_{k}(\Gamma_{0}(N),\psi)_{\mathbb{F}_{p}}$, where the operator $U(p)$ acts on a Fourier series by

[TABLE]

This criterion enables one to verify congruences between modular forms by means of a finite computation.

2.3. Functional Equation and Modular Transformations

The Dedekind eta function satisfies the transformation law

[TABLE]

which plays a central role in the modular transformation behavior of eta-products.

To study modular transformations more precisely, especially under $\mathrm{SL}_{2}(\mathbb{Z})$, we often work with exponential parametrizations of the upper half-plane. Let $w\in\mathbb{C}$ with $\Re(w)>0$, and define

[TABLE]

where $h,k\in\mathbb{N}$, $0\leq h<k$, and $\gcd(h,k)=1$. Let $h^{\prime}$ satisfy $hh^{\prime}\equiv-1\pmod{k}$, and define

[TABLE]

With this notation, the transformation of $\eta$ between the points $\frac{1}{k}(h+iw)$ and $\frac{1}{k}(h^{\prime}+iw^{-1})$ is determined by a certain constant $w_{h,k}$ (known as the $\eta$-multiplier) as follows:

[TABLE]

where

[TABLE]

The above expressions are crucial in analyzing modular transformations of eta-quotients at non-trivial cusps.

3. Modular Construction via Eta-Products Modulo $m$

Let $m\geq 5$ be a prime. Also assume $(m,r_{1})=(m,r_{2})=1$. Define the eta-product

[TABLE]

where $a,b$ are positive integers to be determined. Using the well-known congruence

[TABLE]

it follows that

[TABLE]

Our goal is to select $a$ and $b$ such that the resulting eta-quotient satisfies the conditions in Theorem 2.1. These lead to the congruences

[TABLE]

and

[TABLE]

4. Congruences for the case $r_{1}=r$, $r_{2}=Mr$ with $M$ odd

We note that, for $m$ sufficiently large (so that $bm\geq Mr$), if (3.4) and (3.5) are satisfied, then Theorem 2.1 implies that

[TABLE]

with

[TABLE]

Let $s=\gcd(r,24)$, and write $r=sv$. We take $a=sa_{1}$, $b=sb_{1}$ for some integers $a_{1}$, $b_{1}$. Substituting into (3.4) and (3.5), we obtain:

[TABLE]

Let $d=\gcd(p-M,\tfrac{24}{s})$. We now choose

[TABLE]

so that

[TABLE]

This choice implies that $m^{\prime}\equiv m^{-1}v\equiv mv\pmod{\frac{24}{ds}}$.

We now examine the $q$-expansion of $f_{p,m,r,Mr}(\tau)$. From the definition, we have:

[TABLE]

Applying the $U(m)$ operator and using the congruence $U(m)\equiv T(m)\pmod{m}$, we obtain

[TABLE]

where $T(m)$ denotes the usual Hecke operator acting on $S_{k}(\Gamma_{0}(p),\chi_{p})$.

To isolate a specific subsequence, we extract coefficients for which

[TABLE]

where $d=\gcd(p-M,24)$. This yields:

[TABLE]

Let use denote the right hand side of (4.10) by $g(q)$. This congruence provides the foundation for establishing congruence relations for the coefficients $c_{p,r,Mr}(n)$ modulo $m$ through the modular form $g(q)$.

4.1. Modular Transformation Behavior of Eta Products

Let $\Re(z)>0$ throughout this section. We next determine the growth of the function $g(q)$ appearing on the right-hand side of (4.10) towards all of the cusps. We first deal with the denominator of $g(q)$.

**Case 1: If $p\nmid k$.

**Let $k\in\mathbb{N}$ with $\gcd(h_{0},k)=1$, and suppose $p\nmid k$. Set $h^{\prime}=ph_{0}$ and $w=\frac{z}{p}$, so that $q=e^{\frac{2\pi i}{k}(h^{\prime}+i/w)}$. We choose $h\in\mathbb{Z}$ such that

[TABLE]

Applying the transformation identity (2.19), we obtain

[TABLE]

where

[TABLE]

Similarly, applying (2.19) with $h^{\prime}=h_{0}$ and $w=-iz$ again, we find

[TABLE]

with

[TABLE]

Combining equations (4.13) and (4.15) and taking into account (4.14) and (4.16), we obtain

[TABLE]

for some $f_{n}\in\mathbb{C}$, where $q_{r}:=q^{\frac{1}{r}}$.

**Case 2: $p\mid k$.

**Let $k\in\mathbb{N}$ such that $p\mid k$, and let $H_{0}\in\mathbb{N}$ satisfy $\gcd(H_{0},k)=1$. Set $h^{\prime}=H_{0}$ and $w=z$. Choose $H$ such that

[TABLE]

Then, applying (2.19), we find

[TABLE]

with

[TABLE]

Also,

[TABLE]

with

[TABLE]

Combining (4.19) and (4.21) and plugging in (4.20) and (4.22), we obtain

[TABLE]

for some $l_{n}\in\mathbb{C}$.

Fourier Expansion of the $\eta$-Quotient under Hecke Operator

To determine the growth of the numerator of $g(q)$ on the right-hand side of (4.10) towards the cusp [math], we recall Proposition (2.4)

[TABLE]

where $\kappa$ is defined in (4.2).

Case 1) For $j\geq 1$: In our case, we let $\frac{\tau}{p}=iw=\frac{iz}{p}$, set $h_{j}^{\prime}=pj$, and $k=m$. Then $h_{j}$ satisfies

[TABLE]

For the term $\eta\left(\frac{p(-1/\tau+j)}{m}\right)$ in (2.19), we have:

[TABLE]

where

[TABLE]

Similarly,

[TABLE]

where

[TABLE]

Multiplying powers of (4.27) and (4.29):

[TABLE]

Therefore, combining the above expression together with (4.26) and (4.28), we obtain for $j\geq 1$,

[TABLE]

Recall $q_{N}=e^{\frac{2\pi iz}{N}}$. Summing over $j=1$ to $m-1$ and using von Sterneck’s formula:

[TABLE]

taking into account the fact $\gcd(r,m)=1$ and $pM\not\equiv 1\pmod{m}$, we get

[TABLE]

Case 2: When $j=0$, we will apply the functional equation (2.15) for $\eta$-quotients to obtain (recall that $\tau=iz$)

[TABLE]

For the $V$-operator part, again applying the functional equation (2.15):

[TABLE]

Combining (4.34), (4.35), and (4.33), and keeping in mind that $M$ is odd, we obtain

[TABLE]

Since the numerator on the right hand side of (4.10) belongs to $S_{\kappa}(\Gamma_{0}(p),\chi_{p})$ by (4.1), its Fourier expansion at [math] can only have powers of $q_{p}$, so we can rewrite (4.36), noting (LABEL:on1), as

[TABLE]

for some $t_{n}\in\mathbb{C}$. Here $q_{1}=e^{2\pi i\frac{1}{k}(h^{\prime}+\frac{i}{z})}\to 1$ as $q=e^{2\pi i\frac{1}{k}(h+iz)}\to 0$. Note that, for $m$ sufficiently large, we have

[TABLE]

so (4.37) decays as $q_{1}\to 0$.

Replacing $q$ by $q^{24/d}$ in (4.10), we obtain:

[TABLE]

Combining (4.37), (4.39) and noting (4.10), we conclude:

[TABLE]

where we have used the fact $d=\gcd(p-M,\tfrac{24}{s})=\gcd(pM-1,\tfrac{24}{s})$. Consequently, we will get

[TABLE]

where $g\left(q^{24/d}\right)=\sum_{n\geq 0}u(n)q^{n}$.

Now by Theorem 2.3, the set of primes $\ell$ such that

[TABLE]

has positive density, where $T(\ell)$ denotes the Hecke operator acting on

[TABLE]

where $\kappa$ is defined in (4.2). Moreover, by the theory of Hecke operators, we have:

[TABLE]

Since $u(n)$ vanishes for non-integer $n$, we have $u\left(\frac{n}{\ell}\right)=0$ when $(n,\ell)=1$. Thus,

[TABLE]

Recalling that

[TABLE]

we obtain the final congruence:

[TABLE]

for each integer $n$ with $(n,\ell)=1$.