<s>[INST] Number of elements in discrete $n$-dimensional simplex such that $x_1 \leq \ldots \leq x_n$ For positive integers $n,d$, how many elements are there in the set $S = \{(x_1,\ldots,x_n) \in \mathbb{Z}^n\ |\ 0 \leq x_1 \leq \ldots \leq x_n \wedge \sum_i x_i = d \}$?
I'm hoping that the order constraints on the $x_1,\ldots,x_n$ can be accounted for somehow by "adjusting" the figurate number, which gives the number of elements for an unconstrained discrete simplex. But I'm a bit out of my depth combinatorically.
 [/INST] Introduce change of variable
$$\begin{cases}
x_1 &= y_1\\
x_2 &= y_1 + y_2\\
x_3 &= y_1 + y_2 + y_3\\
&\;\vdots\\
x_n &= y_1 + y_2 + y_3 + \cdots + y_n
\end{cases}$$
We have
$$\begin{array}{c}
0 \le x_1 \le x_2 \le x_n\\
\text{ and }\\
x_1 + x_2 + \cdots + x_n = d
\end{array}
\quad\iff\quad
\begin{array}{c}
y_1, y_2, \ldots, y_n \ge 0\\
\text{ and }\\
ny_1 + (n-1)y_2 + \cdots + y_n = d
\end{array}
$$
This means the number of solutions of $(x_k)$ satisfying LHS is the same as the number of solutions of $(y_k)$ satisfying RHS. Let us denote this number as $p(d,n)$. 
For each solution of $(y_k)$ for the RHS, there is a corresponding partition of integer $d$ into parts whose part not exceeding $n$. i.e.
$$d = \overbrace{n + n + \cdots + n}^{y_1}
    + \overbrace{(n-1) + (n-1) + \cdots + (n-1)}^{y_2}
    + \cdots
    + \overbrace{1 + 1 + \cdots + 1 }^{y_n}$$
This correspondence is one to one. This means $p(d,n)$ equals to the number of ways of expressing integer $d$ as a sum of integers from the set $\{\; 1, 2, \ldots, n \;\}$. For fixed $n$, the generating function of latter is given by:
$$
\text{OCF}_n(t) \stackrel{def}{=} \sum_{d=0}^\infty p(d,n) t^d 
= \prod_{k=1}^n (1 + t^k + t^{2k} + t^{3k} + \cdots )
= \prod_{k=1}^n \frac{1}{1 - t^k}
$$
I'm not aware of any formula of $p(d,n)$ for general $d$ and $n$.
However, for small $n$, you can use these OGFs to derive formula for $p(d,n)$.
For example,


*

*$n = 1$, 


$$\text{OCF}_1(t) = \frac{1}{1-t}\quad\implies\quad
p(d,1) = 1.$$


*

*$n = 2$, 
$$\begin{align}\text{OCF}_2(t) 
&= \frac{1}{(1-t)(1-t^2)} 
= \frac{1}{2(1-t)^2} + \frac{1}{4(1-t)} + \frac{1}{4(1+t)}\\
&=  \frac12 \sum_{d=0}^\infty (d+1)t^d
  + \frac14 \sum_{d=0}^\infty t^d 
  + \frac14 \sum_{d=0}^\infty (-t)^d\\
\implies
p(d,2) &= \frac{d+1}{2} + \frac{1+(-1)^d}{4} = \left\lfloor \frac{d}{2} \right\rfloor + 1
\end{align}
$$

*$n = 3$,
$$\begin{align}
\text{OCF}_3(t) 
&= \frac{1}{(1-t)(1-t^2)(1-t^3)}
= \frac{1/6}{(1-t)^3} + \frac{1/4}{(1-t)^2} + \frac{1/4}{1-t^2} + \frac{1/3}{1-t^3}
\\
&= \frac16\sum_{d=0}^\infty \binom{d+2}{2} t^d + \frac14\sum_{d=0}^\infty (d+1) t^d + \frac14 \sum_{d=0}^\infty t^{2d} + \frac13 \sum_{d=0}^\infty t^{3d}\\
&= \sum_{d=0}^\infty\left( \frac{(d+3)^2}{12} t^d -\frac13 t^d + \frac14 d^{2d} + \frac13 d^{3d}\right)\\
&= \sum_{d=0}^\infty \left(\frac{(d+3)^2}{12} + \epsilon(d)\right) t^d\\
\implies 
p(d,3) &= \frac{(d+3)^2}{12} + \epsilon(d)
\end{align}
$$
where $\epsilon(d)$ takes only the values $-\frac13, -\frac{1}{12}, 0, \frac14$.
Since $p(d,3)$ is an integer and $|\epsilon(d)| < \frac12$, we can further simplify and get 
$$p(d,3) = \left\{ \frac{(d+3)^2}{12} \right\}
\quad\text{ where }
\{ x \} \text{ is the nearest integer to } x.
$$


There are formulas of $p(d,n)$ for other $n$. A good reference should be the book


*

*Integer Partitions, by George E. Andrews, Kimmo Eriksson.


Some of the formula here is copied from this book.
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