Strict Weak Ordering - The Number of Total Preorders

The Number of Total Preorders

The number of distinct total preorders on an n-element set is given by the following sequence (sequence A000670 in OEIS):

Number of n-element binary relations of different types
n all transitive reflexive preorder partial order total preorder total order equivalence relation
0 1 1 1 1 1 1 1 1
1 2 2 1 1 1 1 1 1
2 16 13 4 4 3 3 2 2
3 512 171 64 29 19 13 6 5
4 65536 3994 4096 355 219 75 24 15
OEIS A002416 A006905 A053763 A000798 A001035 A000670 A000142 A000110

These numbers are also called the Fubini numbers or ordered Bell numbers.

As explained above, there is a 1-to-1 correspondence between total preorders and pairs (partition, total order). Thus the number of total preorders is the sum of the number of total orders on every partition. For example:

  • for n = 3:
    • 1 partition of 3, giving 1 total preorder (each element is related to each element)
    • 3 partitions of 2 + 1, giving 3 × 2 = 6 total preorders
    • 1 partition of 1 + 1 + 1, giving 6 total preorders (the total orders)
i.e. together 13 total preorders.
  • for n = 4:
    • 1 partition of 4, giving 1 total preorder (each element is related to each element)
    • 7 partitions with two classes (4 of 3 + 1 and 3 of 2 + 2), giving 7 × 2 = 14 total preorders
    • 6 partitions of 2+1+1, giving 6 × 6 = 36 total preorders
    • 1 partition of 1+1+1+1, giving 24 total preorders (the total orders)
i.e. together 75 total preorders.

Compare the Bell numbers, here 5 and 15: the number of partitions, i.e., the number of equivalence relations.

Read more about this topic:  Strict Weak Ordering

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