Contructing elements of a partially ordered set of increasing integer sequences? Let $\underline{d}=  [\underline{d}_{1},\underline{d}_{2},...,\underline{d}_{n}]$ and $\overline{d}=  [\overline{d}_{1},\overline{d}_{2},...,\overline{d}_{n}]$ where $\underline{d}_{i}\leq \underline{d}_{i+1}\;\forall i=1,..,n-1$ and  $\overline{d}_{i}\leq \overline{d}_{i+1}\;\forall i=1,..,n-1$ be two sequences of increasing integer numbers.
we say $\underline{d} \prec \overline{d}$ iff $\underline{d}_{i} \leq \overline{d}_{i}$ for every $1 \leq i\leq n$, Now Let $[\underline{d},\overline{d}]= \{ d=[d_{1},..,d_{n}]; \underline{d} \prec d \prec\overline{d},\; d_{i}\leq d_{i+1}\;\forall i=1,..,n-1 \}$.
I need to know the number of elements of this partially ordered set, and how can i construct all it's elements or it's maximal chains.
Any hint or help will be greatly appreciated.

I'm facing a problem where I use these sequences.  If this is still of interest, can you tell me if this comes from a bigger problem, and which ?
You can start with $\underline{d}$ and iteratively choose one member $\underline{d}_i$ that you can increase by one to generate $d_1$.  Then you repeat the process until there's no possible choice.  Whatever you do, in the end you'll have increased $\underline{d}_1$ exactly $\overline{d}_1 - \underline{d}_1$ times, $\underline{d}_2$ exactly $\overline{d}_2 - \underline{d}_2$ times, and so on.
So in total, that's $\sum_{i = 1}^n (\overline{d}_i - \underline{d}_i) + 1$.  The $+1$ because we need to include $\underline{d}$ in the set.
