Editing Preorder (section)

==Number of preorders==
{{Number of relations}}

As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). Thus the number of preorders is the sum of the number of partial orders on every partition. For example:
{{unordered list
| for <math>n = 3:</math>
* 1 partition of 3, giving 1 preorder
* 3 partitions of {{nowrap|2 + 1}}, giving <math>3 \times 3 = 9</math> preorders
* 1 partition of {{nowrap|1 + 1 + 1}}, giving 19 preorders

I.e., together, 29 preorders.
| for <math>n = 4:</math>
* 1 partition of 4, giving 1 preorder
* 7 partitions with two classes (4 of {{nowrap|3 + 1}} and 3 of {{nowrap|2 + 2}}), giving <math>7 \times 3 = 21</math> preorders
* 6 partitions of {{nowrap|2 + 1 + 1}}, giving <math>6 \times 19 = 114</math> preorders
* 1 partition of {{nowrap|1 + 1 + 1 + 1}}, giving 219 preorders

I.e., together, 355 preorders.
}}