partitions of positive integer $n$ with respect to a multiset Recently, I think on a new problem related to partitions.
Let $n$ be a non-negative integer and $\mathbb{A}=\{a_1,\ldots,a_k\}$ be a multiset with $k$ 
not necessarily distinct positive integers. We denote by $D(n\mid\mathbb{A})$, the number
of ways to partition $n$ in the form $a_1x_1+\cdots+a_kx_k$, where $x_i$'s are positive integers
and $x_i\leqslant x_{i+1}$ whenever $a_i=a_{i+1}$. 
I would like to calculate generation function of $D(n|A)$. 
 A: Let's try again.  (The original text is gone.  If one would like to waste one's time, it is available through the "edited" link just below the examples.)
Instead of starting with a structureless multiset, let's go with: $I$ is the cardinality of the set of distinct elements in $A$, and $B = \{(b_i, m_i), i \in I\}$ where the $b_i$ are the distinct elements of $A$ and $m_i$ is the multiplicity of $b_i$ in $A$.  So, for example, $\sum_{i \in I} m_i = |A|$.
For each $i$ in $I$, we want a monotonically nondecreasing sequence $n_{i,1} \leq n_{i,2} \leq \cdots \leq n_{i,m_i}$.  We make the change of variables \begin{align*}
    d_{i,1} &= n_{i,1}  \text{,} \\
    d_{i,2} &= n_{i,2} - n_{i,1}  \text{,} \\
    &\vdots  \\
    d_{i,j} &= n_{i,j} - n_{i,j-1}  & (2 \leq j \leq m_i) \text{,} \\
    &\vdots  \\
    d_{i,m_i} &= n_{i,m_i} - n_{i,m_i-1}  \text{.}
\end{align*}
Then the monotonically nondecreasing condition on the $(n_{i,j})_j$ becomes $d_{i,1} \geq 1$ and $d_{i,j} \geq 0$ for $1 \leq j \leq m_i$.  Observe that 
\begin{align*}
\sum_{j=1}^{m_i} n_{i,j} &= (d_{i,1}) + (d_{i,1} + d_{i,2}) + \cdots + (d_{i,1} + d_{i,2} + \cdots + d_{i,m_i})  \\
&= \sum_{j=1}^{m_i} (m_i - j +1) d_{i,j}
\end{align*}
Then $D(n|A)$ is the number of ways of choosing all these $d_{i,j}$ such that \begin{align*}
    n &= \sum_{i \in I} b_i \sum_{j =1}^{m_i} n_{i,j}  \\
      &= \sum_{i \in I} b_i \sum_{j=1}^{m_i} (m_i - j +1) d_{i,j}  \\
      &= \sum_{i \in I} \left( b_i m_i d_{i,1} + \sum_{j=2}^{m_i} b_i (m_i - j +1) d_{i,j} \right)  \text{,}  \\
    d_{i,1} &\geq 1  \qquad (i \in I)  &  \text{, and }  \\
    d_{i,j} &\geq 0  \qquad (i \in I, 2 \leq j \leq m_i)  \text{.}
\end{align*}
The generating function for $D$ is then
$$  \prod_{i \in I} \left( \frac{x^{b_i m_i}}{1-x^{b_i m_i}} \prod_{j = 2}^{m_i} \frac{1}{1-x^{b_i(m_i - j +1)}} \right)  \text{.}  $$
Examples:
\begin{align*}
    \mathrm{gf}(D(n|\{1,2\})) &= x^3 + x^4 + 2 x^5 + 2 x^6 + 3 x^7 + 3 x^8 + 4 x^9 + 4 x^{10} + \cdots  \text{,}  \\
    \mathrm{gf}(D(n|\{1,2,2\})) &= x^5 + x^6 + 2 x^7 + 2 x^8 + 4 x^9 + 4 x^{10} + 6 x^{11} + 6 x^{12} + 9 x^{13} \\
      &\qquad + 9 x^{14} + 12 x^{15} + 12 x^{16} + 16 x^{17} + 16 x^{18} + 20 x^{19} + 20 x^{20} + \cdots  \text{, and} \\
    \mathrm{gf}(D(n|\{1,1,1\})) &= x^3 + x^4 + 2 x^5 + 3 x^6 + 4 x^7 + 5 x^8  + 7 x^9 + 8 x^{10} + \cdots  \text{.}
\end{align*}
Continuing:
We can simplify the gf a bit.  \begin{align*}
\prod_{i \in I} \left( \frac{x^{b_i m_i}}{1-x^{b_i m_i}} \prod_{j = 2}^{m_i} \frac{1}{1-x^{b_i(m_i - j +1)}} \right)
&= \prod_{i \in I}  \frac{x^{b_i m_i}}{1-x^{b_i m_i}} \frac{1}{\prod_{j = 2}^{m_i}1-x^{b_i(m_i - j +1)}}  \\
&= \prod_{i \in I}  \frac{x^{b_i m_i}}{\prod_{j = 1}^{m_i}1-x^{b_i(m_i - j +1)}}  \\
&= \prod_{i \in I}  \prod_{j = 1}^{m_i} \frac{x^{b_i}}{1-x^{b_i(m_i - j +1)}}  \text{.}
\end{align*}
