6. For sets $A$, $B$, and $C$ (not necessarily distinct), the union $A \cup B \cup C=\{1,2,3, \cdots, n\}$. The number of ordered triples of sets $(A, B, C)$ that satisfy this condition is


$6.7^{7}$.
As shown in Figure 7, $A \cup B \cup C$ is divided into 7 non-overlapping parts $X, Y, Z, W, U, V, T$. In this way, each of the elements $1,2,3, \cdots, n$ in $A \cup B \cup C$ has 7 allocation methods, so the total number of allocation schemes for these elements is $7^{n}$. Any 2 different allocation schemes will result in different ordered triples $(A, B, C)$ (since $X, Y, Z, W, U, V, T$ are uniquely determined by the ordered triple $(A, B, C)$), therefore, the number of ordered triples $(A, B, C)$ is equal to the total number of allocation schemes for $1,2,3, \cdots, n$, which is $7^{n}$.