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We have a set of elements $S=\{1,2,3, . . . , N\}$ and family $F$ of $N$ triples of elements in $S$ ($N$ is a multiple of 3). Each element of $S$ appears in exactly three triples. The elements in each triple are distinct. The intersection of two triples $T_1, T_2$ is either empty or has exactly two elements ( $|T_1 \cap T_2|=2$). I guess this should be a well known combinatorial object.

What is the number of such family of triples for each N?

I encountered this object in constructing special instances of exact cover problem. Some of the families have exact cover.

Update: I found a counterexample to the characterization given by Paseman's answer :

Take triples [9,8,6], [2,3,5],[9,7,6],[1,4,5],[1,4,8],[7,3,2],[2,5,1],[6,3,7],[4,8,9]. This is not a disjoint union of tetrahedra.

We have a set of elements $S=\{1,2,3, . . . , N\}$ and family $F$ of $N$ triples of elements in $S$ ($N$ is a multiple of 3). Each element of $S$ appears in exactly three triples. The elements in each triple are distinct. The intersection of two triples $T_1, T_2$ is either empty or has exactly two elements ( $|T_1 \cap T_2|=2$). I guess this should be a well known combinatorial object.

What is the number of such family of triples for each N?

I encountered this object in constructing special instances of exact cover problem. Some of the families have exact cover.

We have a set of elements $S=\{1,2,3, . . . , N\}$ and family $F$ of $N$ triples of elements in $S$ ($N$ is a multiple of 3). Each element of $S$ appears in exactly three triples. The elements in each triple are distinct. The intersection of two triples $T_1, T_2$ is either empty or has exactly two elements ( $|T_1 \cap T_2|=2$). I guess this should be a well known combinatorial object.

What is the number of such family of triples for each N?

I encountered this object in constructing special instances of exact cover problem. Some of the families have exact cover.

Update: I found a counterexample to the characterization given by Paseman's answer :

Take triples [9,8,6], [2,3,5],[9,7,6],[1,4,5],[1,4,8],[7,3,2],[2,5,1],[6,3,7],[4,8,9]

We have a set of elements $S=\{1,2,3, . . . , N\}$ and family $F$ of $N$ triples of elements in $S$ ($N$ is a multiple of 3). Each element of $S$ appears in exactly three triples. The elements in each triple are distinct. The intersection of two triples $T_1, T_2$ is either empty or has exactly two elements ( $|T_1 \cap T_2|=2$). I guess this should be a well known combinatorial object.

What is the number of such family of triples for each N?

I encountered this object in constructing special instances of exact cover problem. Some of the families have exact cover.

Update: I found a counterexample to the characterization given by Paseman's answer :

Take triples [9,8,6], [2,3,5],[9,7,6],[1,4,5],[1,4,8],[7,3,2],[2,5,1],[6,3,7],[4,8,9]. This is not a disjoint union of tetrahedra.

We have a set of elements $S=\{1,2,3, . . . , N\}$ and family $F$ of $N$ triples of elements in $S$ ($N$ is a multiple of 3). Each element of $S$ appears in exactly three triples. The elements in each triple are distinct. The intersection of two triples $T_1, T_2$ is either empty or has exactly two elements ( $|T_1 \cap T_2|=2$). I guess this should be a well known combinatorial object.

What is the number of such family of triples for each N?

I encountered this object in constructing special instances of exact cover problem. Some of the families have exact cover.

We have a set of elements $S=\{1,2,3, . . . , N\}$ and family $F$ of $N$ triples of elements in $S$ ($N$ is a multiple of 3). Each element of $S$ appears in exactly three triples. The elements in each triple are distinct. The intersection of two triples $T_1, T_2$ is either empty or has exactly two elements ( $|T_1 \cap T_2|=2$). I guess this should be a well known combinatorial object.

What is the number of such family of triples for each N?

I encountered this object in constructing special instances of exact cover problem. Some of the families have exact cover.

Update: I found a counterexample to the characterization given by Paseman's answer :

Take triples [9,8,6], [2,3,5],[9,7,6],[1,4,5],[1,4,8],[7,3,2],[2,5,1],[6,3,7],[4,8,9]