Generating Functions for the Number of Permutations with Limited Displacement - Kløve, Torleiv (The Electronic Journal of Combinatorics, 2009)
Let $V(d,n)$ be the number of permutations $p$ of $\{1,2,… ,n\}$ that satisfy $|p_i-i|\le d$ for all $i$. Generating functions for $V(d,n)$, for fixed $d$, are given.