Question 42
In some archipelago, there are $2017$ islands labelled $1, 2, \ldots, 2017$. Two agencies, Red dragon and Blue eye, are planning to establish ship lines between certain islands. Exactly one agency will operate between each two islands, with a ship line from the island labelled with a smaller number to the island labelled with a larger number.
An arrangement of ship lines is good if there are no two islands, labelled $A$ and $B$, where $A \lt B$, such that it is possible to go from $A$ to $B$ in two ways: one, using only the ships owned by Red dragon, and the other, using only ships owned by Blue eye.
Determine the number of good arrangements of ship lines. (Russia 2005)
An arrangement of ship lines is good if there are no two islands, labelled $A$ and $B$, where $A \lt B$, such that it is possible to go from $A$ to $B$ in two ways: one, using only the ships owned by Red dragon, and the other, using only ships owned by Blue eye.
Determine the number of good arrangements of ship lines. (Russia 2005)