## The number of order relations

### Task number: 4148

Determine the number of distinct partial order relations on a set with four elements.

• #### Hint

Proceed systematically according to the number of comparable pairs.

• #### Solution

Linear Arrangement – $$4! = 24$$ options.

One incomparable pair:
between maximum elements – 12 options
between " middle " elements – 12 options
between minimal elements – 12 options

Two incomparable pairs:
two maximum and two minimum elements – 6 options
two maximum elements and one smallest – 24 options
two minimal elements and one largest – 24 options

Three incomparable elements:
with the largest element – 4 options
with the smallest element – 4 options
without the largest and smallest element – $$4! = 24$$ options.

Discontinuous partial arrangements

2 components with two elements – 12 options

2 components, one with one, the other with three elements:
with a linear arrangement on three elements – $$4! = 24$$ options.
with the largest element on the three elements – 12 options
with the smallest element on the three elements – 12 options

3 components – 12 options

4 components – 1 option

The Hasse diagrams for the partial orders are:

• #### Answer

There are 219 partial orders on the four elements, 16 of which are non-isomorphic.