The skew, relative, and classical derangements
A derangement on a set \([n]= \{1,\dots,n\}\) is a permutation \(i_1i_2 \dots i_n\), such that \(i_k\neq k\) for each \(k\). Their number is \(D_n\). A relative derangement \(j_1j_2 \dots j_n\) is a permutation such that \(j_k+ 1\neq j_{k+1}\) for each \(k\). The number of these objects is \(Q_n\). Brualdi showed that \(Q_n=D_n + D_{n-1}\). The reviewed paper gives a bijective proof for this nice correspondence. The main tools are the so-called skew derangements, which are one-to-one maps \(f:X\to Y\) (for \(|X|= |Y|)\) such that for every \(x\in X\) the relation \(f(x)\neq x\) holds.
scientific article; zbMATH DE number 955158