Binary Relations that are Partial Orders I am trying to figure out the relationship between binary relations in a set and partial orders. Any thoughts?
 A: Kaa1el is correct in the comments: this is OEIS A$001035$, and the desired term is $19$. Probably the easiest way to compute it by hand is to list the possible Hasse diagrams and count the possible relations for each. Here are the five possible Hasse diagrams:
  *    
  |  
  *         *   *            *             *  
  |          \ /            / \            |  
  *           *            *   *           *   *           *   *   *

There are $3!=6$ distinct partial orders with the first diagram, one for each permutation of $a,b$, and $c$ from top to bottom. A partial order on $\{a,b,c\}$ with the second diagram is completely determined when we know which element is at the bottom, so it accounts for just $3$ partial orders. Similarly, the third Hasse diagram accounts for $3$. The fourth again accounts for $6$: there are $3$ ways to choose the odd element and then $2$ ways to order the other two. Finally, the last Hasse diagram corresponds to just one partial order on $\{a,b,c\}$, equality. The total is then $6+3+3+6+1=19$.
