On the Enumeration and Congruences for m-ary Partitions

TL;DR

This paper provides a new representation for the number of m-ary partitions of an integer and explores their congruences, connecting combinatorial structures with base m representations and extending previous results.

Contribution

It introduces a novel j-fold summation formula for m-ary partitions and establishes a correspondence with base m representations, extending known congruence characterizations.

Findings

Derived a j-fold summation representation for b_m(n)

Connected m-ary partitions to base m representations

Extended congruence results for c_m(n) and b_m(n)

Abstract

Let $m\ge 2$ be a fixed positive integer. Suppose that $m^j \leq n< m^{j+1}$ is a positive integer for some $j\ge 0$. Denote $b_{m}(n)$ the number of $m$-ary partitions of $n$, where each part of the partition is a power of $m$. In this paper, we show that $b_m(n)$ can be represented as a $j$-fold summation by constructing a one-to-one correspondence between the $m$-ary partitions and a special class of integer sequences rely only on the base $m$ representation of $n$. It directly reduces to Andrews, Fraenkel and Sellers' characterization of the values $b_{m}(mn)$ modulo $m$. Moreover, denote $c_{m}(n)$ the number of $m$-ary partitions of $n$ without gaps, wherein if $m^i$ is the largest part, then $m^k$ for each $0\leq k<i$ also appears as a part. We also obtain an enumeration formula for $c_m(n)$ which leads to an alternative representation for the congruences of $c_m(mn)$ due to Andrews, Fraenkel, and Sellers.

Tables

Table 1. Table 2.1: The correspondence between λ ∈ ℬ 4 ​ ( 36 ) 𝜆 subscript ℬ 4 36 \lambda\in\mathcal{B}_{4}(36) and β ∈ 𝒮 4 ​ ( 36 ) 𝛽 subscript 𝒮 4 36 \beta\in\mathcal{S}_{4}(36)

$\lambda$	$(2,1,0)$	$(2,0,4)$	$(1,5,0)$	$(1,4,4)$	$(1,3,8)$	$(1,2,12)$	$(1,1,16)$	$(1,0,20)$	$(0,9,0)$
$\beta$	$(0,0)$	$(0,1)$	$(1,0)$	$(1,1)$	$(1,2)$	$(1,3)$	$(1,4)$	$(1,5)$	$(2,0)$
$\lambda$	$(0,8,4)$	$(0,7,8)$	$(0,6,12)$	$(0,5,16)$	$(0,4,20)$	$(0,3,24)$	$(0,2,28)$	$(0,1,32)$	$(0,0,36)$
$\beta$	$(2,1)$	$(2,2)$	$(2,3)$	$(2,4)$	$(2,5)$	$(2,6)$	$(2,7)$	$(2,8)$	$(2,9)$

Equations

$$n=a_{\ell }{m}^{\ell }+a_{\ell -1}{m}^{\ell -1}+\cdots +a_0,$$

$$B_m(q)=\sum _{n=0}^{\infty }{b}_m(n)q^n=\prod _{k=0}^{\infty }\frac{1}{1-q^{m^k}}.$$

$$b_2(2^{2k+2}n)\equiv b_2(2^{2k}n)(\mathrm{mod}{2}^{3k+2}),$$

$$b_2(2^{2k+1}n)\equiv b_2(2^{2k-1}n)(\mathrm{mod}{2}^{3k}),$$

$$n={\alpha }_j{m}^j+{\alpha }_{j-1}{m}^{j-1}+\cdots +{\alpha }_{1}m+{\alpha }_0,$$

$$r_m(n)=({\alpha }_j,{\alpha }_{j-1},\dots ,{\alpha }_1,{\alpha }_0).$$

$$b_m(mn)\equiv \prod _{i=0}^j({\alpha }_i+1)(\mathrm{mod}m).$$

$${\mathcal{S}}_m(n)=\{({\beta }_j,{\beta }_{j-1},\dots ,{\beta }_1)|0\le {\beta }_j\le {\alpha }_j\text{mbox}and0\le {\beta }_t\le {\alpha }_t+m{\beta }_{t+1}\text{mbox}for1\le t\le j-1\}.$$

$$b_m(n)=\sum _{k_j=0}^{{\alpha }_j}\sum _{k_{j-1}=0}^{{\alpha }_{j-1}+m{k}_j}\dots \sum _{k_1=0}^{{\alpha }_1+m{k}_2}1.$$

$$b_m(mn)=\sum _{k_j=0}^{{\alpha }_j}\sum _{k_{j-1}=0}^{{\alpha }_{j-1}+m{k}_j}\dots \sum _{k_1=0}^{{\alpha }_1+m{k}_2}\sum _{k_0=0}^{{\alpha }_0+m{k}_1}1.$$

$$c_m(n)=1+\sum _{r=1}^j\sum _{k_r={\chi }_r}^{\lfloor \frac{n}{m^r}\rfloor -1}\dots \sum _{k_1={\chi }_1}^{{\alpha }_1-1+m{k}_2}1,$$

$$\chi_{i}=\left\{\begin{array}[]{ll}0,&\hbox{if $\alpha_{i-1}>0$,}\\[5.0pt] 1,&\hbox{if $\alpha_{i-1}=0$.}\end{array}\right.$$

$$c_m(mn)$$

$${\mathcal{S}}_m(n)=\{({\beta }_j,{\beta }_{j-1},\dots ,{\beta }_1)|0\le {\beta }_j\le {\alpha }_j\text{mbox}and0\le {\beta }_t\le {\alpha }_t+m{\beta }_{t+1}\text{mbox}for1\le t\le j-1\},$$

$$r_m(n)-\lambda =({\beta }_j,{\beta }_{j-1},\dots ,{\beta }_1),$$

$${\beta }_i=\sum _{k=i}^j{m}^{k-i}({\alpha }_k-{\lambda }_k)$$

$${\beta }_j={\alpha }_j-{\lambda }_j,$$

$${\beta }_t={\alpha }_t-{\lambda }_t+m{\beta }_{t+1},$$

$${\lambda }_j{m}^j+{\lambda }_{j-1}{m}^{j-1}+\cdots +{\lambda }_0={\alpha }_j{m}^j+{\alpha }_{j-1}{m}^{j-1}+\cdots +{\alpha }_0,$$

$${\lambda }_j{m}^j+{\lambda }_{j-1}{m}^{j-1}+\cdots +{\lambda }_t{m}^t\le {\alpha }_j{m}^j+{\alpha }_{j-1}{m}^{j-1}+\cdots +{\alpha }_0.$$

$$\Big{(}(\lambda_{t}-\alpha_{t})+\sum_{k=1}^{j-t}(\lambda_{t+k}-\alpha_{t+k})m^{k}\Big{)}m^{t}\leq\alpha_{t-1}m^{t-1}+\alpha_{t-2}m^{t-2}+\cdots+\alpha_{0}.$$

$${\alpha }_{t-1}{m}^{t-1}+{\alpha }_{t-2}{m}^{t-2}+\cdots +{\alpha }_0<m^t,$$

$$\Big{(}(\lambda_{t}-\alpha_{t})+\sum_{k=1}^{j-t}(\lambda_{t+k}-\alpha_{t+k})m^{k}\Big{)}m^{t}<m^{t}.$$

$${\lambda }_t-{\alpha }_t+\sum _{k=1}^{j-t}({\lambda }_{t+k}-{\alpha }_{t+k})m^k\le 0$$

$${\lambda }_t\le {\alpha }_t+\sum _{k=1}^{j-t}({\alpha }_{t+k}-{\lambda }_{t+k})m^k={\alpha }_t+m{\beta }_{t+1}.$$

$${\lambda }^{\prime }=({\alpha }_j-{\beta }_j,{\alpha }_{j-1}-{\beta }_{j-1}+m{\beta }_j,\dots ,{\alpha }_1-{\beta }_1+m{\beta }_2,{\alpha }_0+m{\beta }_1)$$

$$({\alpha }_j$$

$$f({\alpha }_j,{\alpha }_{j-1},\dots ,{\alpha }_1,{\alpha }_0)=\sum _{k_j=0}^{{\alpha }_j}\sum _{k_{j-1}=0}^{{\alpha }_{j-1}+m{k}_j}\dots \sum _{k_2=0}^{{\alpha }_2+m{k}_3}\sum _{k_1=0}^{{\alpha }_1+m{k}_2}1.$$

$${\mathcal{S}}_m(n)=\{({\beta }_i,{\beta }_{i-1},\dots ,{\beta }_1)|0\le {\beta }_i\le {\alpha }_i,0\le {\beta }_t\le {\alpha }_t+m{\beta }_{t+1},1\le t\le i-1\}.$$

$$\mathcal{S}_{\beta_{i}}=\big{\{}(\beta_{i-1},\beta_{i-2},\ldots,\beta_{1})\,|\,0\leq\beta_{t}\leq\alpha_{t}+m\beta_{t+1},1\leq t\leq i-1\big{\}},$$

$${\mathcal{S}}_m(n)=\bigcup _{{\beta }_i=0}^{{\alpha }_i}{\mathcal{S}}_{{\beta }_i}.$$

Full text

On the Enumeration and Congruences for $m$-ary Partitions

Lisa Hui Sun1 and Mingzhi Zhang2

Center for Combinatorics, LPMC

Nankai University, Tianjin 300071, P. R. China

1sunhui@nankai.edu.cn, 2mingzhi@mail.nankai.edu.cn

Abstract. Let $m\geq 2$ be a fixed integer. Suppose that $n$ is a positive integer such that $m^{j}\leq n<m^{j+1}$ for some integer $j\geq 0$. Denote $b_{m}(n)$ the number of $m$-ary partitions of $n$, where each part of the partition is a power of $m$. In this paper, we show that $b_{m}(n)$ can be represented as a $j$-fold summation by constructing a one-to-one correspondence between the $m$-ary partitions and a special class of integer sequences relying only on the base $m$ representation of $n$. It directly reduces to Andrews, Fraenkel and Sellers’ characterization of the values $b_{m}(mn)$ modulo $m$. Moreover, denote $c_{m}(n)$ the number of $m$-ary partitions of $n$ without gaps, wherein if $m^{i}$ is the largest part, then $m^{k}$ for each $0\leq k<i$ also appears as a part. We also obtain an enumeration formula for $c_{m}(n)$ which leads to an alternative representation for the congruences of $c_{m}(mn)$ modulo $m$ due to Andrews, Fraenkel and Sellers.

Keywords: $m$-ary partition, base $m$ representation, congruence

AMS Classification: 05A17, 11P83

1 Introduction

The arithmetic properties for partition functions have been extensively studied since the discoveries of Ramanujan [12]. In this paper, we are mainly concerned with the enumeration of $m$-ary partitions which leads to the congruence properties given by Andrews, Fraenkel and Sellers [3, 4].

Let $m\geq 2$ be a fixed integer. An $m$-ary partition of a positive integer $n$ is a partition of $n$ such that each part is a power of $m$. The number of $m$-ary partitions of $n$ is denoted by $b_{m}(n)$. For example, there are five $3$-ary partitions for $n=10$:

9+1, 3+3+3+1, 3+3+1+1+1+1,  3+1+1+1+1+1+1+1,  1+1+1+1+1+1+1+1+1+1.

Thus, $b_{3}(10)=5$. Denote an $m$-ary partition of $n$ by a sequence $\lambda=(a_{\ell},a_{\ell-1},\ldots,a_{0})$ such that

[TABLE]

where $a_{\ell}>0$ and $a_{i}\geq 0$ for $0\leq i\leq\ell-1$. Denote the set of all the $m$-ary partitions of $n$ by $\mathcal{B}_{m}(n)$. It is known that the generating function of $b_{m}(n)$ is given by

[TABLE]

For the case $m=2$, Churchhouse [6] conjectured the following congruences for the binary partition function $b_{2}(n)$:

[TABLE]

where $n,k\geq 1$. The conjecture was first proved by Rødseth [13] and further studied by Hirschhorn and Loxton [9]. Later, it was extended to $m$-ary partitions by Andrews [1], Gupta [8], and Rødseth and Sellers [14].

Throughout this paper, without specification, we set $n$ to be a positive integer such that $m^{j}\leq n<m^{j+1}$ for some integer $j\geq 0$. Recall that the base $m$ representation of $n$ is the unique expression of $n$ which can be written as follows

[TABLE]

where $\alpha_{j}>0$ and $0\leq\alpha_{i}\leq m-1$ for $0\leq i\leq j-1$. Denote the base $m$ representation of $n$ by

[TABLE]

Based on the base $m$ representation of $n$, Andrews, Fraenkel and Sellers [3, Theorem 1] provided the following modulo $m$ characterization of $b_{m}(mn)$:

[TABLE]

In this paper, by establishing a bijection between the set $\mathcal{B}_{m}(n)$ of $m$-ary partitions of $n$ and the set of integer sequences given in the following theorem, we derive a $j$-fold summation formula for $b_{m}(n)$. It will directly lead to Andrews, Fraenkel and Sellers’ congruence (1.5).

Theorem 1.1**.**

There is a one-to-one correspondence between the set $\mathcal{B}_{m}(n)$ of $m$-ary partitions of $n$ and the following set of integer sequences

[TABLE]

Based on the above bijection, we provide a combinatorial approach to derive the following $j$-fold summation formula for $b_{m}(n)$.

Theorem 1.2**.**

Let $r_{m}(n)$=$(\alpha_{j},\alpha_{j-1},\ldots,\alpha_{1},\alpha_{0})$ be the base $m$ representation of $n$. We have

[TABLE]

Obviously, $b_{m}(n)=1$ when $j=0$.

Notice that if $r_{m}(n)$=$(\alpha_{j},\alpha_{j-1},\cdots,\alpha_{1},\alpha_{0})$, then $r_{m}(mn)$=$(\alpha_{j},\alpha_{j-1},\cdots,\alpha_{1},\alpha_{0},0)$. Thus the above theorem leads to that

[TABLE]

By taking modulo $m$ on both sides of the above equation, it directly reduces to Andrews, Fraenkel and Sellers’ congruence (1.5).

We also consider the cases for the $m$-ary partitions without gaps, wherein if $m^{i}$ is the largest part, then $m^{k}$ for each $0\leq k<i$ also appears as a part. The related works on such restricted $m$-ary partitions can be found in [2, 4, 11]. Moreover, in [5, 7, 10, 15], a general class of non-squashing partitions was introduced and studied, which contains $m$-ary partitions as a special case.

Let $c_{m}(n)$ denote the number of $m$-ary partitions without gaps of $n$. Based on the bijection given in Theorem 1.1, we also obtain the following enumeration formula for $c_{m}(n)$.

Theorem 1.3**.**

Let $r_{m}(n)=(\alpha_{j},\alpha_{j-1},\ldots,\alpha_{1},\alpha_{0})$ be the base $m$ representation of $n$. We have

[TABLE]

where for $1\leq i\leq r$,

[TABLE]

Applying formula (1.7), we obtain the following congruence property of $c_{m}(mn)$, which reveals the results given by Andrews, Fraenkel and Sellers [4, Theorem 2.1].

Theorem 1.4**.**

Let $\chi_{i}$ be defined by (1.8) for $1\leq i\leq r$, then we have

[TABLE]

2 The enumeration formula for $b_{m}(n)$

In this section, we provide a bijection between the set $\mathcal{B}_{m}(n)$ of $m$-ary partitions of $n$ and the set

[TABLE]

which relies only on the base $m$ representation of $n$. It will lead to the enumeration formula (1.6) for the $m$-ary partitions.

To this end, we first define the following subtraction between the base $m$ representation of $n$ and an ordinary $m$-ary partition of $n$.

Definition 2.1**.**

Let $r_{m}(n)=(\alpha_{j},\alpha_{j-1},\ldots,\alpha_{0})$ be the base $m$ representation of $n$ and $\lambda=(\lambda_{\ell},\lambda_{\ell-1},\ldots,\lambda_{0})$ be an $m$-ary partition of $n$. Then subtracting $\lambda$ from $r_{m}(n)$ is given as follows

[TABLE]

where for $1\leq i\leq j$,

[TABLE]

provided that $\lambda_{k}=0$ for $\ell<k\leq j$.

We further show that the subtraction (2.1) defined above gives a bijection between $\mathcal{B}_{m}(n)$ and $\mathcal{S}_{m}(n)$.

Theorem 2.2**.**

Let $r_{m}(n)=(\alpha_{j},\alpha_{j-1},\ldots,\alpha_{0})$ be the base $m$ representation of $n$ and $\lambda=(\lambda_{\ell},\lambda_{\ell-1},\ldots,\lambda_{0})$ be an arbitrary $m$-ary partition of $n$. Define a map $\varphi$ from $\mathcal{B}_{m}(n)$ to $\mathcal{S}_{m}(n)$ by $\varphi(\lambda)=r_{m}(n)-\lambda$. Then $\varphi$ is a bijection between $\mathcal{B}_{m}(n)$ and $\mathcal{S}_{m}(n)$.

Proof. Denote $\beta=\varphi(\lambda)=(\beta_{j},\beta_{j-1},\ldots,\beta_{1})$. First, we proceed to show that $\beta\in\mathcal{S}_{m}(n)$ and thereby $\varphi$ is well defined. Following Definition 2.1, it is easy to see that

[TABLE]

where $1\leq t\leq j-1$ and $\lambda_{k}=0$ for $\ell<k\leq j$. Since $\lambda_{k}\geq 0$ for $0\leq k\leq j$, we see that $\beta_{j}\leq\alpha_{j}$ and $\beta_{t}\leq\alpha_{t}+m\beta_{t+1}$ for $1\leq t\leq j-1$.

It is obvious that $\lambda_{j}\leq\alpha_{j}$, so that $\beta_{j}\geq 0$. From the fact that

[TABLE]

we are led to that for $1\leq t\leq j-1$,

[TABLE]

Hence we obtain that

[TABLE]

Since $(\alpha_{j},\alpha_{j-1},\ldots,\alpha_{0})$ is the base $m$ representation of $n$, it is obvious that

[TABLE]

which implies that

[TABLE]

Note that $\lambda_{k}$ and $\alpha_{k}$ are all integers for $0\leq k\leq j$, it follows that

[TABLE]

and thereby

[TABLE]

By (2.3), it directly leads to that $\beta_{t}\geq 0$ for $1\leq t\leq j-1$. Thus $\beta\in\mathcal{S}_{m}(n)$ and $\varphi$ is well defined.

To prove that $\varphi$ is a bijection, it is sufficient to show that there exists the inverse map of $\varphi$. For a given $\beta=(\beta_{j},\beta_{j-1},\ldots,\beta_{1})\in\mathcal{S}_{m}(n)$, let $\varphi^{-1}(\beta)$ be given by computing

[TABLE]

and then deleting the preceding zeros. From the definition of $\mathcal{S}_{m}(n)$, we see that each element of $\lambda^{\prime}$ is nonnegative. Furthermore, it is easy to see that

[TABLE]

which implies that $\varphi^{-1}(\beta)$ is an $m$-ary partition of $n$ and thereby $\varphi^{-1}(\beta)\in\mathcal{B}_{m}(n)$. It completes the proof of the bijection.

For example, let $m=4$ and $n=36$, then the base $4$ representation of $36$ is $r_{4}(36)=(2,1,0)$. The correspondence between all the $4$-ary partitions of $36$ and the integer sequences belonging to $\mathcal{S}_{4}(36)$ can be seen in Table 2.1.

The above theorem directly leads to that $b_{m}(n)=|\mathcal{B}_{m}(n)|=|\mathcal{S}_{m}(n)|$. By studying the recursive properties of the sequences in $\mathcal{S}_{m}(n)$, we obtain the $j$-fold summation formula (1.6) of $b_{m}(n)$. Now we give the detailed proof of Theorem 1.2.

Proof of Theorem 1.2. Denote the summation on the right hand side of (1.6) by

[TABLE]

We prove the theorem by induction. When $j=0$, $r_{m}(n)=(\alpha_{0})$ with $0\leq\alpha_{0}\leq m-1$. It is obvious that $b_{m}(n)=f(\alpha_{0})=1$.

Suppose that (1.6) holds for $j=i-1$. When $j=i$, we have $r_{m}(n)=(\alpha_{i},\alpha_{i-1},\ldots,\alpha_{1},\alpha_{0})$. By Theorem 2.2, it implies that $b_{m}(n)=|\mathcal{S}_{m}(n)|$ where

[TABLE]

For a fixed $\beta_{i}$ with $0\leq\beta_{i}\leq\alpha_{i}$, let us consider the subset of $\mathcal{S}_{m}(n)$ with $\beta_{i}$ being the first entry. By deleting $\beta_{i}$ in these sequences, it is easy to see that this subset is bijective with the following set

[TABLE]

and therefore

[TABLE]

By induction, we obtain that the cardinality of the set $\mathcal{S}_{\beta_{i}}$ is

[TABLE]

Then by summing the above equation for $\beta_{i}$ from [math] to $\alpha_{i}$, we obtain

[TABLE]

which completes the proof.

Note that the $j$-fold summation formula (1.6) also can be derived from the generating function (1.1) of $b_{m}(n)$. Moreover, by setting $M=\{m,m,\ldots\}$ in the summation given by Folsom, Homma, Ryu and Tong [7, Theorem 1.5], it reduces to another $j$-fold summation expression for $b_{m}(n)$.

3 The $m$-ary partitions without gaps

In this section, based on the bijection given in Theorem 2.2, we derive an enumeration formula for the $m$-ary partitions without gaps. Denote $c_{m}(n)$ the number of this restricted $m$-ary partitions of $n$. We also obtain an alternative expression for the congruence properties of $c_{m}(mn)$ given by Andrews, Fraenkel and Sellers [4, Theorem 2.1].

Recall that by using the base $m$ representation of $n$ in the following form

[TABLE]

where $1\leq\alpha_{\ell}<m$ and $0\leq\alpha_{i}<m$ for $i>\ell$, Andrews, Fraenkel and Sellers obtained the following result.

Theorem 3.1** (Andrews, Fraenkel and Sellers [4, Theorem 2.1]).**

• (1)

If $\ell$ is even, then

[TABLE]

• (2)

If $\ell$ is odd, then

[TABLE]

Denote the floor function of a real number $a$ by $\lfloor a\rfloor$, which is the largest integer less than or equal to $a$. To derive our expression of the congruences (3.1) and (3.2), first let us show how to derive the enumeration formula (1.7) for $c_{m}(n)$ as given in Theorem 1.3.

Proof of Theorem 1.3. Denote the set of all the $m$-ary partitions without gaps of $n$ by $\mathcal{G}_{m}(n)$. We claim that for any $\lambda\in\mathcal{G}_{m}(n)$, it can be written as

[TABLE]

where $0\leq r\leq j$ and $\beta_{i}$ are integers such that

[TABLE]

Specially, when $r=0$, there is a unique $m$-ary partition without gaps, say, $\lambda=(n)$ which consists of $n$ ones. For $r>1$, as an example, we consider $m=4$ and $n=73$, then $r_{4}(73)=(1,0,2,1)$. When $r=2$, we can obtain that $(3,6,1)$ is a $4$-ary partition without gaps which can be represented as \big{(}\left\lfloor\frac{73}{4^{2}}\right\rfloor-1,2-0+4\times 1,1+4\times 0\big{)}.

Recall that

[TABLE]

For $1\leq r\leq j$, it follows that

[TABLE]

which certifies that $\lambda\in\mathcal{G}_{m}(n)$. For a fixed integer $r$ such that $0\leq r\leq j$, by the bijection given in Theorem 2.2, we get

[TABLE]

Note that for any $\lambda\in\mathcal{G}_{m}(n)$ with the given $r$, the first $j-r$ elements in $\varphi(\lambda)$ are the same, which only depend on $\alpha_{r+1},\ldots,\alpha_{j}$. Then by deleting these terms, we find that the set of $m$-ary partitions without gaps $\mathcal{G}_{m}(n)$ is in one-to-one correspondence with the following set of integer sequences

[TABLE]

It indicates that $c_{m}(n)=|\mathcal{R}_{m}(n)|$.

From the conditions (3.3), we see that if $\alpha_{0}=0$, then $\beta_{1}>0$, which means $\beta_{1}$ starts from 1. If $\alpha_{0}>0$, then $\beta_{1}\geq 0$, which means $\beta_{1}$ starts from 0. For both cases, we denote $\beta_{1}$ starting from $\chi_{1}$, which is defined by (1.8). By $\alpha_{1}-\beta_{1}+m\beta_{2}>0$ we have $\beta_{1}<\alpha_{1}+m\beta_{2}$. Thereby we see that $\beta_{1}$ ranges from $\chi_{1}$ to $\alpha_{1}+m\beta_{2}-1$. By similar arguments applying to $\beta_{i}$ for $2\leq i\leq r$, we have

[TABLE]

where for $1\leq i\leq j$,

[TABLE]

This completes the proof.

Noting that $r_{m}(mn)=(\alpha_{j},\alpha_{j-1},\ldots,\alpha_{0},0)$, then by applying the above result we have

[TABLE]

By taking modulo $m$ on both sides of the above identity, we directly obtain the congruence property (1.9), namely,

[TABLE]

As an example, let $m=5$ and $n=485=3\cdot 5^{3}+4\cdot 5^{2}+2\cdot 5$. Then $j=3$ and the base $5$ representation of $485$ is $r_{5}(485)=(\alpha_{3},\alpha_{2},\alpha_{1},\alpha_{0})=(3,4,2,0)$. Therefore

[TABLE]

and

[TABLE]

In fact, we have $c_{5}(5\cdot 485)=230358\equiv 3\pmod{5}$, which coincides with the above result.

To conclude this paper, we remark that the congruence (3) for $c_{m}(mn)$ is equivalent to Theorem 3.1 due to Andrews, Fraenkel and Sellers [4].

Proof of Theorem 3.1. Let $r_{m}(n)$=$(\alpha_{j},\alpha_{j-1},\ldots,\alpha_{1},\alpha_{0})$ be the base $m$ representation of $n$.

Following Lemma 2.9 of [4], we see that $c_{m}(m^{3}n)\equiv c_{m}(mn)\pmod{m}$ for all $n\geq 0$. Thereby to prove Theorem 3.1, it is sufficient to show the cases that $\ell=0$ and $\ell=1$, which correspond to $\alpha_{0}>0$ and $\alpha_{0}=0$ ($\alpha_{1}>0$), respectively.

If $\alpha_{0}>0$, then $\chi_{1}=0$. It leads to that

[TABLE]

We further consider the values of $\alpha_{1},\alpha_{2},\cdots,\alpha_{j}$. If $\alpha_{i}>0$ for $i\geq 1$, then $\chi_{i}=0$ for $2\leq i\leq j$. Thus (3.5) turns to be

[TABLE]

where $\alpha_{i}=0$ for $i>j$. Otherwise, suppose that $\alpha_{k}\ (1\leq k\leq j)$ is the first zero in the sequence $\alpha_{1},\alpha_{2},\cdots,$ then $\chi_{i}=0$ for $1\leq i\leq k$. Noting that $\alpha_{k}=0$, we obtain

[TABLE]

where $\alpha_{i}=0$ for $i>j$. Therefore, (3.5) leads to that

[TABLE]

and both cases coincide with (3.1) with $\ell=0$.

If $\alpha_{0}=0$ and $\alpha_{1}>0$, following the same procedure, we obtain that

[TABLE]

which coincides with (3.2) with $\ell=1$. This completes the proof.

Acknowledgments.

This work was supported by the National Science Foundation of China, and the Natural Science Foundation of Tianjin, China.