(Library chain). On each request the i th of n possible books is the one chosen with probability pi . To make it quicker to find the book the next time, the librarian moves the book to the left end of the shelf. Define the state at any time to be the sequence of books we see as we examine the shelf from left to right. Since all the books are distinct this list is a permutation of the set f1; 2; : : : ng, i.e., each number is listed exactly once. Show that .i1; : : : ; in/ D pi1 pi2 1 [1] pi1 pi3 1 [1] pi1 [1] pi2 pin 1 [1] pi1 [1] pin[1]1 is a stationary distribution.