Binomial coefficients and anti-exceedances of even permutations: A combinatorial proof
For a permutation \(\pi\in S_ n\), the anti-exceedance \(\text{AX}(\pi)\) is the number of integers \(i\) which are transformed into a smaller integer by \(\pi\). For the alternating group \(A_ n\) define \(P_{n,k}=|\{\pi\in A_ n:\text{AX}(\pi)=k\}|\) and \(D_{n,k}=|\{\pi\in S_ n-A_ n:\text{AX}(\pi)=k\}|\). It is well known that \(P_{n,k}+D_{n,k}=\) the Eulerian number \(A_{n,k}\). The author gives a bijective proof for \(P_{n,k}-D_{n,k}=(-1)^{n- k}{n-1\choose k-1}\).
scientific article; zbMATH DE number 217952