Crown

Example

Let $n \ge 3$. The crown $S_n$ is the partially ordered set on $2n$ elements $a_1, \ldots, a_n, b_1, \ldots, b_n$ whose only strict relations are $a_i < b_j$ for all $i \ne j$. It is also called the standard example of dimension $n$, because its order dimension is $n$. Some authors use the name crown for a different poset of height $2$, whose comparability graph is a cycle of length $2n$.

Lower bound. For each $i$, the elements $a_i$ and $b_i$ are incomparable, so a realizer needs a linear extension $L$ with $b_i$ before $a_i$. Suppose one linear extension $L$ has $b_i$ before $a_i$ and $b_j$ before $a_j$ with $i \ne j$. Since $a_i < b_j$ and $a_j < b_i$, $L$ would contain the cycle $a_i, b_j, a_j, b_i, a_i$, which is impossible. So each linear extension reverses at most one pair, and at least $n$ linear extensions are needed.

Upper bound. For each $i$, let $L_i$ be the following order. First come the $a_j$ with $j \ne i$, in increasing order of $j$, then $b_i$, then $a_i$. Last come the $b_j$ with $j \ne i$, in increasing order of $j$. Each $L_i$ is a linear extension, and every incomparable pair appears in both relative orders among $L_1, \ldots, L_n$, so these $n$ orders form a realizer.