June 04, 2025
We prove new formulas for \(\operatorname{p}_d(n)\), the number of \(d\)-ary partitions of \(n\), and, also, for \(P_d(n)\), its polynomial part.
Given a partition \(\lambda=(\lambda_1,\ldots,\lambda_{\ell})\), its associated \(j\)-th symmetric elementary partition, \(\operatorname{pre}_{j}(\lambda)\), is the partition whose parts are \(\{\lambda_{i_1}\cdots\lambda_{i_j}\;:\;1\leq i_1 < \cdots < i_j\leq \ell\}\). We prove that if \(\lambda\) and \(\mu\) are two \(d\)-ary partitions of length \(\ell\) such that \(\operatorname{pre}_j(\lambda)=\operatorname{pre}_j(\mu)\) and \(\lambda_{i_1}\cdots \lambda_{i_j} = \mu_{i_1}\cdots \mu_{i_j}\), for all \(1\leq i_1 < \cdots < i_j\leq \ell\), then \(\lambda=\mu\).
Let \(n\) be a positive integer. We denote \([n]=\{1,2,\ldots,n\}\). A partition of \(n\) is a non-increasing sequence of positive integers \(\lambda_i\) whose sum equals \(n\). We define \(p(n)\) as the number of partitions of \(n\) and we define \(p(0) = 1\). We denote \(\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_{\ell})\) with \(\lambda_1\geq \lambda_2\geq \cdots \geq \lambda_{\ell} \geq 1\) and \(|\lambda|:=\lambda_1+\cdots+\lambda_{\ell}=n\). We refer to \(|\lambda|\) as the size of \(\lambda\) and the numbers \(\lambda_i\) as parts of \(\lambda\). The number \(\ell(\lambda)=\ell\) is the number of parts of \(\lambda\) and it is called the length of \(\lambda\). For more on the theory of partitions, we refer the reader to [1].
Let \(d\geq 2\) be an integer. A partition \(\lambda=(\lambda_1,\ldots,\lambda_{\ell})\) is called \(d\)-ary, if all \(\lambda_i\)’s are powers of \(d\). A \(2\)-ary partition is called binary. In Proposition 1 we establish a natural bijection between the set of all integer partition and the set of \(d\)-ary partitions, which conserves the length (but not the size).
In Theorem 1 we give a new formula for \(p_d(n)\), the number of \(d\)-ary partitions of \(n\), using the fact that a \(d\)-ary partition is a partition with the parts in \(\{1,d,d^2,d^3,\ldots\}\). In Theorem 1, we give a new formula for \(W_j(d,n)\)’s, the Sylvester waves of \(p_d(n)\). Also, in Theorem 1 and Theorem 1 we give new formulas for \(P_d(n)=W_1(d,n)\), the polynomial part of \(p_d(n)\). In Example 1, we illustrate these results.
Now, let \(K\) be an arbitrary field and \(S=K[x_1,\ldots,x_{\ell}]\) be the ring of polynomials over \(K\) in \(\ell\) indeterminates. We recall that the \(j^{th}\) elementary symmetric polynomial of \(S\) is \[e_j(x_1,\ldots,x_{\ell})=\sum_{1\leq i_1 < i_2 <\ldots <i_j\leq \ell}x_{i_1}x_{i_2}\cdots x_{i_j},\text{ where }1\leq j\leq \ell.\] Also, we define \(e_0(x_1,\ldots,x_{\ell})=1\) and \(e_j(x_1,\ldots,x_{\ell})=0\) for \(j>\ell\).
Given a partition \(\lambda\), we have \(e_j(\lambda)=0\) if \(\ell(\lambda)<j\) and \[e_j(\lambda)=\sum_{1\leq i_1 < i_2 <\ldots <i_j\leq \ell(\lambda)}\lambda_{i_1}\lambda_{i_2}\cdots \lambda_{i_j},\text{ if }1\leq j\leq \ell(\lambda).\] In [2], [3], Ballantine et al. introduced the following definition. Given a partition \(\lambda\), the partition \(\operatorname{pre}_{j}(\lambda)\) is the partition whose parts are \[\{\lambda_{i_1}\cdots\lambda_{i_j}\;:\;1\leq i_1 < i_2 < \cdots < i_j\leq \ell(\lambda)\},\] and they called it an elementary symmetric partition. Note that \(\operatorname{pre}_1(\lambda)=\lambda\), but \(\operatorname{pre}_j(\lambda)\neq \lambda\), for \(j\geq 2\). For example, if \(\lambda=(3,2,1,1)\), then \(\operatorname{pre}_{2}(\lambda)=(6,3,3,2,2,1)\).
A natural question to ask is the following: If \(\lambda\) and \(\mu\) are two partitions such that \(\operatorname{pre}_j(\lambda)=\operatorname{pre}_j(\mu)\) then is it true that \(\lambda=\mu\)? Only the following cases are known in literature: (i) \(j=2\) and \(m(\lambda),m(\mu)\leq 3\), see [3] and (ii) \(j=2\) and \(\lambda\) and \(\mu\) are binary partitions; see [3]. In Theorem 1 we prove that if \(\lambda\) and \(\mu\) are two \(d\)-ary partitions of length \(\ell\) such that \(\operatorname{pre}_j(\lambda)=\operatorname{pre}_j(\mu)\) and \(\lambda_{i_1}\cdots \lambda_{i_j} = \mu_{i_1}\cdots \mu_{i_j}\), for all \(1\leq i_1 < \cdots < i_j\leq \ell\), where \(1\leq j\leq \ell-1\), then \(\lambda=\mu\).
Let \(\mathbf{a} := (a_1, a_2, \ldots , a_r)\) be a sequence of positive integers, where \(r \geq 1\). Let \(\lambda\) be a partition. We say that \(\lambda\) has parts in \(\mathbf{a}\) if \(\lambda_i \in \{a_1,\ldots,a_r\}\) for all \(1\leq i\leq \ell(\lambda)\).
The restricted partition function associated to \(\mathbf{a}\) is \(p_{\mathbf{a}}: \mathbb{N} \to \mathbb{N}\), \(p_{\mathbf{a}}(n) :=\) the number of integer solutions \((x_1, \ldots, x_r)\) of \(\sum_{i=1}^r a_ix_i = n\) with \(x_i \geq 0\). In other words, \(p_{\mathbf{a}}(n)\) counts the number of partitions of \(n\) with parts in \(\mathbf{a}\). Note that the generating function of \(p_{\mathbf{a}}(n)\) is \[\label{gen} \sum_{n=0}^{\infty}p_{\mathbf{a}}(n)z^n= \frac{1}{(1-z^{a_1})\cdots(1-z^{a_r})}.\tag{1}\] Let \(D\) be a common multiple of \(a_1\), \(a_2,\ldots,a_r\). Bell [4] proved that \(p_{\mathbf{a}}(n)\) is a quasi-polynomial of degree \(k-1\), with the period \(D\), that is \[\label{quasi} p_{\mathbf{a}}(n)=d_{\mathbf{a},k-1}(n)n^{k-1}+\cdots+d_{\mathbf{a},1}(n)n+d_{\mathbf{a},0}(n),\tag{2}\] where \(d_{\mathbf{a},m}(n+D)=d_{\mathbf{a},m}(n)\) for \(0\leq m\leq k-1\) and \(n\geq 0\), and \(d_{\mathbf{a},k-1}(n)\) is not identically zero.
Sylvester [5],[6] decomposed the restricted partition in a sum of “waves”: \[\label{wave} p_{\mathbf{a}}(n)=\sum_{j\geq 1} W_{j}(n,\mathbf{a}),\tag{3}\] where the sum is taken over all distinct divisors \(j\) of the components of \(\mathbf{a}\) and showed that for each such \(j\), \(W_j(n,\mathbf{a})\) is the coefficient of \(t^{-1}\) in \[\sum_{0 \leq \nu <j,\; \operatorname{gcd}(\nu,j)=1 } \frac{\rho_j^{-\nu n} e^{nt}}{(1-\rho_j^{\nu a_1}e^{-a_1t})\cdots (1-\rho_j^{\nu a_k}e^{-a_kt}) },\] where \(\rho_j=e^{\frac{2\pi i}{j}}\) and \(\operatorname{gcd}(0,0)=1\) by convention. Note that \(W_{j}(n,\mathbf{a})\)’s are quasi-polynomials of period \(j\). Also, \(W_1(n,\mathbf{a})\) is called the polynomial part of \(p_{\mathbf{a}}(n)\) and it is denoted by \(P_{\mathbf{a}}(n)\).
Theorem 1. ([7])We have \[p_{\mathbf{a}}(n) = \frac{1}{(r-1)!} \sum_{\substack{0\leq j_1\leq \frac{D}{a_1}-1,\ldots, 0\leq j_r\leq \frac{D}{a_r}-1 \\ a_1j_1+\cdots+a_rj_r \equiv n (\bmod D)}} \prod_{\ell=1}^{r-1} \left(\frac{n-a_1j_{1}- \cdots -a_rj_r}{D}+\ell \right).\]
The unsigned Stirling numbers are defined by \[\binom{n+r-1}{r-1}=\frac{1}{n(r-1)!}n^{(r)}=\frac{1}{(r-1)!}\left(\genfrac{[}{]}{0pt}{}{r}{r}n^{r-1} + \cdots \genfrac{[}{]}{0pt}{}{r}{2}n + \genfrac{[}{]}{0pt}{}{r}{1}\right).\]
Theorem 1. ([8])For any positive integer \(j\) with \(j|a_i\) for some \(1\leq i\leq r\), we have that \[\begin{align} W_{j}(n,\mathbf{a}) = \frac{1}{D(r-1)!} \sum_{m=1}^r \sum_{\ell=1}^{j} \rho_j^{\ell} \sum_{k=m-1}^{r-1} \genfrac{[}{]}{0pt}{}{r}{k+1} (-1)^{k-m+1} \binom{k}{m-1} \times \\ \times \sum_{\substack{0\leq j_1\leq \frac{D}{a_1}-1,\ldots, 0\leq j_r\leq \frac{D}{a_r}-1 \\ a_1j_1+\cdots+a_rj_r \equiv \ell (\bmod j)}} D^{-k} (a_1j_1+\cdots+a_rj_r)^{k-m+1} n^{m-1}. \end{align}\]
Theorem 1. ([7])For the polynomial part \(P_{\mathbf{a}}(n)\) of the quasi-polynomial \(p_{\mathbf{a}}(n)\) we have \[P_{\mathbf{a}}(n) = \frac{1}{D(r-1)!} \sum_{0\leq j_1\leq \frac{D}{a_1}-1,\ldots, 0\leq j_r\leq \frac{D}{a_r}-1} \prod_{\ell=1}^{r-1} \left(\frac{n-a_1j_{1}- \cdots -a_rj_r}{D}+\ell \right).\]
The Bernoulli numbers \(B_{\ell}\)’s are defined by \(\frac{t}{e^t-1}=\sum_{\ell=0}^{\infty}\frac{t^{\ell}}{\ell !}B_{\ell}\). We have \(B_0=1\), \(B_1 = -\frac{1}{2}\), \(B_2=\frac{1}{6}\), \(B_4=-\frac{1}{30}\) and \(B_n=0\) is \(n\) is odd and \(n\geq 1\).
Theorem 1. ([7] or [9])The polynomial part of \(p_{\mathbf{a}}(n)\) is \[P_{\mathbf{a}}(n) := \frac{1}{a_1\cdots a_r}\sum_{u=0}^{r-1}\frac{(-1)^u}{(r-1-u)!}\sum_{i_1+\cdots+i_r=u} \frac{B_{i_1}\cdots B_{i_r}}{i_1!\cdots i_r!}a_1^{i_1}\cdots a_r^{i_r} n^{r-1-u}.\]
We fix \(d\geq 2\) an integer. We denote \(\mathcal{P}\), the set of integer partitions, and \(\mathcal{P}_d\), the set of \(d\)-ary partitions. Given a positive integer \(n\), we denote \(p_d(n)\), the number of \(d\)-ary partitions of \(n\).
Definition 1. Let \(\lambda=(\lambda_1,\ldots,\lambda_{\ell})\in\mathcal{P}\) be a partition. The \(d\)-exponential of \(\lambda\) is the \(d\)-ary partition: \[\operatorname{Exp}_d(\lambda):=(d^{\lambda_1-1},\ldots,d^{\lambda_{\ell}-1}).\]
Definition 1. Let \(\lambda=(\lambda_1,\ldots,\lambda_{\ell})\in\mathcal{P}_d\) be a \(d\)-ary partition. The \(d\)-logarithm of \(\lambda\) is the partition: \[\operatorname{Log}_d(\lambda):=(\log_d(\lambda_1)+1,\ldots,\log_d(\lambda_{\ell})+1).\]
Proposition 1. The maps \(\operatorname{Exp}_d:\mathcal{P} \to \mathcal{P}_d\) and \(\operatorname{Log}_d:\mathcal{P}_d \to \mathcal{P}\) are bijective and inverse of each other.
Proof. Let \(\lambda=(\lambda_1,\ldots,\lambda_{\ell})\in\mathcal{P}\). We have \(\operatorname{Exp}_d(\lambda)=(d^{\lambda_1-1},\ldots,d^{\lambda_{\ell}-1})\). Since \[\log_d(d^{\lambda_i-1})+1=\lambda_i-1+1=\lambda_i\text{ for all }1\leq i\leq \ell,\] it follows that \(\operatorname{Log}_d(\operatorname{Exp}_d(\lambda))=\lambda\). Similary, if \(\mu\in \mathcal{P}_d\) is a \(d\)-ary partition, then it is easy to see that \(\operatorname{Exp}_d(\operatorname{Log}_d(\mu))=\mu\). Hence, the proof is complete. ◻
Lemma 1. Let \(n\) and \(k\) be two positive integers such that \(n<d^{k+1}\). The number of \(d\)-ary partitions of \(n\) is \[p_d(n)=p_{(1,d,\ldots,d^{k})}(n).\] In particular, the polynomial part of \(p_d(n)\) is \(P_d(n)=P_{(1,d,\ldots,d^{k})}(n)\).
Proof. Let \(\lambda=(\lambda_1,\ldots,\lambda_{\ell})\) be a \(d\)-ary partition of \(n\), that is \(n=|\lambda|\). It follows that \(\lambda_i=d^{c_i}\) with \(0\leq c_i\) and \(d^{c_i}\leq n\) for all \(1\leq i\leq \ell\). Since \(\lambda_1 = d^{c_1} \leq |\lambda| < d^{k+1}\) and \(\lambda_1\geq \lambda_2\geq \cdots \geq \lambda_{\ell}\), it follows that \[k\geq c_1\geq c_2\geq \cdots \geq c_{\ell}\geq 0,\] and, therefore, \(\lambda\) is a partition with parts in \((1,d,\ldots,d^k)\). On the other hand, any partition with parts in \((1,d,\ldots,d^k)\) is a \(d\)-ary partition. Hence, the proof is complete. ◻
Theorem 1. Let \(n\) and \(k\) be two positive integers such that \(n<d^{k+1}\). The number of \(d\)-ary partitions of \(n\) is \[p_d(n)=\frac{1}{k!} \sum_{\substack{0\leq j_1\leq d^{k}-1,\; 0\leq j_2\leq d^{k-1}-1,\; \ldots, 0\leq j_{k} \leq d-1 \\ j_1+j_2d+\cdots+j_kd^{k-1} \equiv n (\bmod d^k)}} \prod_{\ell=1}^{k} \left(\frac{n-j_{1}- j_2 d- \cdots -j_k d^{k-1}}{d^k}+\ell \right).\]
Proof. According to Lemma 1, we have \(p_d(n)=p_{(1,d,\ldots,d^k)}(n)\), where \(k=\lfloor \log_d(n) \rfloor\). Hence, the conclusion follows from Theorem 1, taking \(r=k+1\) and \(D=\operatorname{lcm}(1,d,\ldots,d^k)=d^k\). ◻
From Lemma 1 and 3 we can write \[p_d(n)=\sum_{j\geq 1}W_j(d,n),\text{ where }W_j(d,n)=W_j(n,(1,d,\ldots,d^k)),\] and \(k=\lfloor \log_d(n) \rfloor\). In particular, the polynomial part of \(p_d(n)\) is \[P_d(n)=W_1(d,n).\]
Theorem 1. Let \(n\) and \(k\) be two positive integers such that \(n<d^{k+1}\). We have that \[\begin{align} W_j(d,n)=\frac{1}{k!d^k} \sum_{m=1}^{k+1} \sum_{\ell=1}^{j} \rho_j^{\ell} \sum_{s=m-1}^{k} \genfrac{[}{]}{0pt}{}{k+1}{s+1} (-1)^{s-m+1} \binom{s}{m-1} \times \\ \times \sum_{\substack{0\leq j_1\leq d^k-1,\ldots, 0\leq j_k\leq d-1 \\ j_1+dj_2+\cdots+d^{k-1}j_{k-1} \equiv \ell (\bmod j)}} d^{-ks} (j_1+dj_2+\cdots+d^{k-1}j_{k-1})^{s-m+1} n^{m-1}. \end{align}\]
Theorem 1. Let \(n\) and \(k\) be two positive integers such that \(n<d^{k+1}\). The polynomial part of \(p_d(n)\) is \[P_d(n)=\frac{1}{k!d^k} \sum_{0\leq j_1\leq d^{k}-1,\; 0\leq j_2\leq d^{k-1}-1,\; \ldots, 0\leq j_{k} \leq d-1} \prod_{\ell=1}^{k} \left(\frac{n-j_{1}- j_2 d- \cdots -j_k d^{k-1}}{d^k}+\ell \right).\]
Theorem 1. Let \(n\) and \(k\) be two positive integers such that \(n<d^{k+1}\). The polynomial part of \(p_d(n)\) is \[P_d(n)= \frac{1}{d^{\frac{k(k+1)}{2}}} \sum_{u=0}^{k} \frac{(-1)^u}{(k-u)!}\sum_{i_1+\cdots+i_{k+1}=u} \frac{B_{i_1}\cdots B_{i_{k+1}}}{i_1!\cdots i_{k+1}!}d^{i_2+2i_3+\cdots+ki_{k+1}} n^{k-u}.\]
Example 1. (1) Let \(n=8\) and \(d=3\). Since \(n<d^{1+1}\), Theorem 1 implies \[p_3(8)=\frac{1}{1!}\sum_{0\leq j_1\leq 2,\;j_1\equiv 8(\bmod 3)} \left(\frac{8-j_1}{3}+1 \right) = \frac{8-2}{3}+1 = 3.\] Also, from Theorem 1 it follows that the polynomial part of \(p_3(8)\) is \[P_3(8)=\frac{1}{1!\cdot 3^1}\sum_{j_1=0}^2 \left(\frac{8-j_1}{3}+1 \right) = \frac{1}{9}\sum_{j_1=0}^2(11-j_1)=\frac{11+10+9}{9}=\frac{10}{3}.\] (2) Let \(n=20\) and \(d=3\). Since \(n<d^{2+1}\), Theorem 1 implies \[p_3(20)=\frac{1}{162}\sum_{\substack{0\leq j_1\leq 8,\;0\leq j_2\leq 2\\ j_1+3j_2\equiv 20(\bmod 9)}} (29-j_1-3j_2)(38-j_1-3j_2).\] Since the set of pairs \((j_1,j_2)\) which satisfy the above conditions is \(\{(2,0),(8,1),(5,2)\}\), it follows that \(p_3(20)=\frac{1}{162}(28\cdot 36 + 18\cdot 27 + 18\cdot 27)=12\).
Given \(n\geq 2\) an integer, we denote by \(\{e_1,\ldots,e_n\}\), the standard basis of the vector space \(\mathbb{R}^n\), i.e. \(e_i\) is the vector with \(1\) in the \(i\)-th position and zeros everywhere else.
Let \(1\leq j\leq n-1\) be an integer. We consider the vectors: \[c_i=\begin{cases} e_1+e_2+\cdots+e_j,& i=1 \\ e_1+e_2+\cdots+e_{j+1}-e_{i-1},&2\leq i\leq j+1 \\ e_{i-j+1}+e_{i-j+2}+\cdots+e_{i},& j+2\leq i\leq n\end{cases}\] Let \(C\) be the \(n\times n\) matrix whose columns are \(c_1,c_2,\ldots,c_n\).
To better illustrate the structure of the matrix \(C\), we present the case \(n=6\) and \(j=3\): \[C = \begin{pmatrix} 1 & 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}.\]
Lemma 1. With the above notations, we have that \(\det(C)=j\).
Proof. From the definition of \(C\), we easily note that \(\det(C)=\det(A)\), where \[A=\begin{pmatrix} 1 & 0 & 1 & \cdots & 1 \\ 1 & 1 & 0 & \cdots & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \cdots & 0 \\ 0 & 1 & 1 & \cdots & 1 \end{pmatrix}\] is a \((j+1)\times(j+1)\) circulant matrix with the associated polynomial \[f(x)=1+x+x^2+\cdots+x^{j-1}.\] For more details on circulant matrices, we refer the reader to [10].
Let \(\omega=e^{\frac{2\pi i}{j+1}}\) be a primitive \((j+1)\)-th root of unity. Using a basic result on circulant matrices, we have that \[\det(A)=\prod_{k=0}^{j} f(\omega^k).\] It is clear that \(f(\omega^0)=f(1)=j\). On the other hand, for \(1\leq k\leq j\), we have that \[f(\omega^k)=1+\omega^k+\cdots+\omega^{k(j-1)}=-\omega^{kj}.\] Therefore, it follows that \[\det(A)=(-1)^{j}j\omega^{\frac{j^2(j+1)}{2}}.\] If \(j\) is even, then \[\omega^{\frac{j^2(j+1)}{2}}=(\omega^{j+1})^{\frac{j^2}{2}}=1^{\frac{j^2}{2}} =1.\] On the other hand, if \(j\) is odd, then \[\omega^{\frac{j^2(j+1)}{2}}=(\omega^{\frac{(j+1)}{2}})^{j^2}=(-1)^{j^2}=-1.\] Hence, in both cases, we have that \(\det(A)=j\). Thus, the proof is complete. ◻
Theorem 1. Let \(\lambda\) and \(\mu\) be two \(d\)-ary partitions with \(\ell\) parts and let \(1\leq j\leq \ell-1\) be an integer. If \(\operatorname{pre}_j(\lambda)=\operatorname{pre}_j(\mu)\) and and \(\lambda_{i_1}\cdots \lambda_{i_j} = \mu_{i_1}\cdots \mu_{i_j}\), for all \(1\leq i_1 < \cdots < i_j\leq \ell\), then \(\lambda=\mu\).
Proof. Since \(\lambda\) is a \(d\)-ary partition, it follows that \(\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_{\ell})\) such that \(\lambda_i=d^{c_i}\), for all \(1\leq i\leq \ell\), and \(c_1\geq c_2\geq \cdots \geq c_{\ell}\). Similarly, \(\mu=(\mu_1,\ldots,\mu_{\ell})\) with \(\mu_i=d^{c'_i}\), for all \(1\leq i\leq \ell\), and \(c_1\geq c_2\geq \cdots \geq c_{\ell}\).
From the definition, \(\operatorname{pre}_j(\lambda)\) is the partition whose parts are: \[\{ d^{c_{i_1}+c_{i_2}+\cdots+c_{i_j}}\;:\;1\leq i_1 < i_2 < \cdots < i_j\leq \ell \}.\] Similarly, \(\operatorname{pre}_j(\mu)\) is the partition whose parts are: \[\{ d^{c'_{i_1}+c'_{i_2}+\cdots+c'_{i_j}}\;:\;1\leq i_1 < i_2 < \cdots < i_j\leq \ell \}.\] Since \(\operatorname{pre}_j(\lambda)=\operatorname{pre}_j(\mu)\) and \(\lambda_{i_1}\cdots \lambda_{i_j} = \mu_{i_1}\cdots \mu_{i_j}\), for all \(1\leq i_1 < \cdots < i_j\leq \ell\), it follows that \[c'_{i_1}+c'_{i_2}+\cdots+c'_{i_j} = c_{i_1}+c_{i_2}+\cdots+c_{i_j},\text{for all }1\leq i_1 < i_2 < \cdots < i_j\leq \ell.\] For convenience, we denote \[c_{i_1,\ldots,i_j}:=c_{i_1}+c_{i_2}+\cdots+c_{i_j},\text{for all }1\leq i_1 < i_2 < \cdots < i_j\leq \ell.\] From Proposition 1, in order to prove that \(\lambda=\mu\), it suffices to show that \((c_1,\ldots,c_{\ell})=(c'_1,\ldots,c'_{\ell})\). In order to do that, it is enough to prove that the linear system \[\label{sistem} \begin{cases} x_{i_1}+x_{i_2}+\cdots+x_{i_j} = c_{i_1,\ldots,i_j} \end{cases},\text{where }1\leq i_1 < i_2 < \cdots < i_j\leq \ell,\tag{4}\] has a unique solution. Since \((c_1,\ldots,c_{\ell})\) is already a solution of 4 , it is enough to prove that the matrix associated to 4 has the rank \(n\). We consider the following subsystem of 4 : \[\label{ssistem} \begin{cases} x_1+x_2+\cdots+x_j = c_{1,2,\ldots,j} \\ x_2+x_3+\cdots+x_{j+1}=c_{2,\ldots,j+1} \\ x_1+x_3+\cdots+x_{j+1}=c_{1,3,\ldots,j+1} \\ \vdots \\ x_1+\cdots+x_{j-1}+x_{j+1}=c_{1,\ldots,j-1,j+1}\\ x_3+x_4+\cdots+x_{j+2} = c_{3,\ldots,j+2}\\ x_4+x_5+\cdots+x_{j+3} = c_{4,\ldots,j+3}\\ \vdots \\ x_{\ell-j+1}+\cdots+x_{\ell}=c_{\ell-j+1,\ldots,\ell} \end{cases}.\tag{5}\] Note that the matrix associated to 5 is \(C^T\), where \(C\) was defined at the beginning of this section.
According to Lemma 1 we have \(\det(C^T)=\det(C)=j\neq 0\). Hence, 5 has a unique solution. Thus 4 has also a unique solution, as required. ◻
Let \(n\geq 1\) and \(d\geq 2\) be two integers. We proved new formulas for \(\operatorname{p}_d(n)\), the number of \(d\)-ary partitions of \(n\), and, also, for \(P_d(n)\), its polynomial part.
Given \(\lambda\) a partition of length \(\ell\) and \(1\leq j\leq \ell-1\), we denote \(\operatorname{pre}_j(\lambda)\), its associated \(j\)-th elementary symmetric partition; see [2], [3]. Given \(\lambda\) and \(\mu\) two \(d\)-ary partitions of length \(\ell\) and \(1\leq j\leq \ell-1\), we proved that if \(\operatorname{pre}_j(\lambda)=\operatorname{pre}_j(\mu)\) and \(\lambda_{i_1}\cdots \lambda_{i_j} = \mu_{i_1}\cdots \mu_{i_j}\), for all \(1\leq i_1 < \cdots < i_j\leq \ell\), then \(\lambda=\mu\), thus giving a partial positive answer to a problem raised in [2].