Stationary position of a transposed distinguished book (${\pi }_i\propto {\left(\frac{1-p}{p(N-1)}\right)}^{i-1}$)

If a distinguished book is selected with probability $p$ and each other book with probability $q=(1-p)/(N-1)$, swapping the selected book with its left neighbor makes the distinguished position a birth-death chain with left rate $p$ and right rate $q$. Stationary flow balance gives successive probabilities in ratio $q/p$, yielding a truncated geometric distribution; the uniform case occurs when all books are equally likely.