# Tag Archives: Family of sets

## Enumeration of Antimatroids, part II

As explained before, we would like to extend the following sequence A119770 (don’t click, it will spoil all the fun!):
1, 1, 3, 22, 485, 59386, ...

## Enumeration of Antimatroids, part I

Recently I found an interesting problem: enumeration of possible antimatroids over finite set.

From combinatorial point of view, an antimatroid is a family of sets, with special properties.
Following the wikipedia definition, $F$ is an antimatroid iff:

• $F$ is closed under unions, that is: $\forall_{u,v \in F} u \cup v \in F$
• $F$ is a nonempty accesible: $\forall_{u \in F} ((u = \emptyset) \vee (\exists_{a \in u} u \setminus \{a\} \in F))$

For example, following family is an antimatroid:
$\{\{\},\{0\},\{2\},\{0,1\},\{0,2\},\{1,2\},\{0,1,2\}\}$
From now, I will denote the largest set, being also a superset of every other one in the family, as $S$ (so in previous example, $S=\{0,1,2\}$).

Just as matroids are generalization of antichains, antimatroids are generalization of chains.