Assume there is a sequence $\{b_n\}$ which satisfies the conditions below:
$1. b_1 + b_2 + \cdots + b_{n} = m, m \ge n, m \in N, n \in N$
$2. b_1 \ge b_2 \ge \cdots \ge b_{n} \ge 1, b_i \in N$
How many different sequences $\{b_n\}$ are there? And if I want to find every specific sequence which satisfies these 2 conditions, are there any convenient programming algorithms that perform efficiently?
PS: My problem is a little bit different from the Wikipedia page of Partition. I need to separate an integer $m$ to $n$ parts with order (from big one to small one).
Consider the Wikipedia page, below there is a reccurence relation. For partitions satisfying only the first condition you can go as follows (maybe you get inspiration):
Consider the series expansion of $$ \frac{z}{1-z} = \sum_{k=1}^{\infty} z^k, $$ when we take the $n$-th power we get $$ \frac{z^n}{(1-z)^n} = \sum_{k_1=1}^{\infty} z^{k_1} \cdots \sum_{k_n=1}^{\infty} z^{k_n}. $$ Denote by $C^m$ the number of possibilities such that $n$ numbers ($\geq 1$) add up to $m$. Then we have $$ \frac{z^n}{(1-z)^n} = \sum_{m=1}^{\infty} C^m z^m.$$ On the other hand the power expansion of $\frac{z^n}{(1-z)^n}$ around $z=0$ can easily seen to be $$ \frac{z^n}{(1-z)^n} = \sum_{m=0}^{\infty} \begin{pmatrix} n+m-1 \\ m \end{pmatrix} z^{m+n}=\sum_{m=n}^{\infty} \begin{pmatrix} m-1 \\ m-n \end{pmatrix} z^{m}. $$ Equating coefficient gives that $C^m = \begin{pmatrix} m-1 \\ m-n \end{pmatrix}$.