Generalized Eulerian Numbers and Directed Friends-and-seats Graphs

Let $A(n,m)$ denote the Eulerian numbers, which count the number of permutations on $[n]$ with exactly m descents, or, due to the Foata transform, the number of permutations on $[n]$ with exactly m excedances. Friends-and-seats graphs, also known as friends-and-strangers graphs, are a seemingly unrelated recent construction in graph theory. In this paper, we introduce directed friends-and-seats graphs and establish a connection between these graphs and a generalization of the Eulerian numbers. We use this connection to reprove and extend a Worpitzky-like identity on generalized Eulerian numbers.