Refinements of (\(n,m\))-Dyck paths
A \((n,m)\)-Dyck path is a lattice path in \(\mathbb{Z}\times\mathbb{Z}\) using up \((1,1)\) and down \((1,-1)\) steps that go from the origin to the point \((2n,0)\) and it contains exactly \(m\) up steps under the \(x\)-axis. The classical Chung-Feller theorem tells us that the number of \((n,m)\)-Dyck paths is \(\frac{1}{n+1}\binom{2n}{n}\) the \(n\)-th Catalan number. In this paper, the authors considered refinements of \((n,m)\)-Dyck paths by using four parameters: peaks, valleys, double-descents and double-ascents. In particular, they showed that the number of \((n,m)\)-Dyck paths having exactly \(k\)-peaks is given by \(\frac{2(n+2)}{n(n-1)}\binom{n}{k-1}\binom{n}{k}\) for all \(m=1,2,\ldots,n-1\) and independent of \(m\). Moreover, they found that the number of \((n,m)\)-Dyck paths having exactly \(k\) double-ascents (double-descents) is given by \(\frac{1}{k+1}\binom{n-1}{k}\binom{n}{k}\).
scientific article; zbMATH DE number 5817921