I'm looking at a problem where I have a space of states $\mathcal{S} = \{0,1,\ldots,m\}^n$ where $m$ and $n$ are given integers. Consider an elementwise ordering between two states $s,s'\in\mathcal{S}$, denoted by $\succeq$, where $s\succeq s'$ if $s_i\ge s_i'$ for all $i=1,\ldots,n$. For example $s=(0,1,1)\succeq s'=(0,1,0)$ but $s=(0,1,1)\not\succeq s'=(1,1,0)$, where in the later case $s$ and $s'$ are not orderable, termed unorderable.
Goal: To determine the number of unorderable pairs of states, with respect to the elementwise partial order, for a given $m$ and $n$.
Example 1: Consider $m=1$ and $n=2$, then $\mathcal{S}=\{0,1\}^2$ and the unorderable pairs of states are simply $(0,1)\not\succeq(1,0)$ (all other pairs of states are orderable). Note that I treat the case $(1,0)\not\succeq(0,1)$ as equivalent to $(0,1)\not\succeq(1,0)$.
Example 2: Consider $m=1$ and $n=3$, then $\mathcal{S}=\{0,1\}^3$ and the unorderable pairs of states are \begin{align*} (0,0,1)&\not\succeq(0,1,0)\\ (0,0,1)&\not\succeq(1,0,0)\\ (0,0,1)&\not\succeq(1,1,0)\\ (0,1,0)&\not\succeq(1,0,0)\\ (0,1,0)&\not\succeq(1,0,1)\\ (0,1,1)&\not\succeq(1,0,0)\\ (0,1,1)&\not\succeq(1,0,1)\\ (0,1,1)&\not\succeq(1,1,0)\\ (1,1,0)&\not\succeq(1,0,1)\\ \end{align*} How can one characterize the set of unorderable states for larger $m$ and $n$?
Instead of counting incomparable pairs of states, count the comparable ones.
How many pairs of states are there where $a\preceq b$? For each coordinate, $i$, the pair $(a_i,b_i)$ must satisfy $a_i\le b_i$. There are $\binom{m+1}2+m+1=(m+1)(m+2)/2$ ways to choose $(a_i,b_i)$. Therefore, there are $[(m+1)(m+2)/2]^n$ pairs of states where $a\preceq b$. Similarly, there are $[(m+1)(m+2)/2]^n$ pairs of states where $a\succeq b$. In total, the number of pairs of comparable states is $$ [(m+1)(m+2)/2]^n + [(m+1)(m+2)/2]^n - (m+1)^n $$ Note that we subtracted $(m+1)^n$ at the end, since the pairs where $a=b$ were double counted.
Finally, subtract the above quantity from $(m+1)^{2n}$, the total number of pairs of states, and divide by two to turn ordered pairs into unordered pairs.