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• A facet $F$ of a simplicial complex is called a leaf if either $F$ is the only facet of $\Delta$, or there exists a facet $G \in \Delta \setminus {F}$ such that
  

  $$F \cap \left( \bigcup_{H \in \Delta \setminus {F}} H \right) \subseteq G.$$

• A connected simplicial complex $\Delta$ is a tree if every nonempty subcollection (that is, a subcomplex of $\Delta$ whose facets are also facets of $\Delta$) has a leaf.

In the language of simplicial complexes, simplicial trees are equivalent to totally balanced hypergraphs [see Theorem 3.2 of [Herzog, Jürgen; Hibi, Takayuki; Trung, Ngô Viêt; Zheng, Xinxian, Standard graded vertex cover algebras, cycles and leaves, Trans. Am. Math. Soc. 360, No. 12, 6231-6249 (2008)].

A hypergraph is called totally balanced if every cycle $C$ in $H$ of length at least 3 (not necessarily odd-length) has an edge containing at least three vertices of $C$. A cycle in a hypergraph is a sequence of distinct vertices and hyperedges $(v_1, e_1, v_2, e_2, \dots, v_k, e_k, v_{k+1} = v_1),$ where every vertex $v_i$ is contained in both $e_{i-1}$ and $e_i$. The number $k$ is called the length of the cycle.

My question. Is there any code in SageMath to generate all simplicial trees (equivalently, totally balanced hypergraphs) on $n$ vertices, having as output the list of hyperedges, and ideally also some visual representation?

• A facet $F$ of a simplicial complex is called a leaf if either $F$ is the only facet of $\Delta$, or there exists a facet $G \in \Delta \setminus {F}$ such that
  

  $$F \cap \left( \bigcup_{H \in \Delta \setminus {F}} H \right) \subseteq G.$$

• A connected simplicial complex $\Delta$ is a tree if every nonempty subcollection (that is, a subcomplex of $\Delta$ whose facets are also facets of $\Delta$) has a leaf.

In the language of simplicial complexes, simplicial trees are equivalent to totally balanced hypergraphs [see Theorem 3.2 of [Herzog, Jürgen; Hibi, Takayuki; Trung, Ngô Viêt; Zheng, Xinxian, Standard graded vertex cover algebras, cycles and leaves, Trans. Am. Math. Soc. 360, No. 12, 6231-6249 (2008)].

A hypergraph is called totally balanced if every cycle $C$ in $H$ of length at least 3 (not necessarily odd-length) has an edge containing at least three vertices of $C$. A cycle in a hypergraph is a sequence of distinct vertices and hyperedges $(v_1, e_1, v_2, e_2, \dots, v_k, e_k, v_{k+1} = v_1),$ where every vertex $v_i$ is contained in both $e_{i-1}$ and $e_i$. The number $k$ is called the length of the cycle.

My question. Is there any code in SageMath to generate all simplicial trees (equivalently, totally balanced hypergraphs) on $n$ vertices, having as output the list of hyperedges, and ideally also some visual representation?

A hypergraph is called totally balanced if every cycle $C$ in $H$ of length at least 3 (not necessarily odd-length) has an edge containing at least three vertices of $C$. A cycle in a hypergraph is a sequence of distinct vertices and hyperedges $(v_1, e_1, v_2, e_2, \dots, v_k, e_k, v_{k+1} = v_1),$ where every vertex $v_i$ is contained in both $e_{i-1}$ and $e_i$. The number $k$ is called the length of the cycle.

My question. Is there any code in SageMath to generate all simplicial trees (equivalently, totally balanced hypergraphs) hypergraphs on $n$ vertices, having as output the list of hyperedges, and ideally also some visual representation?