  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.
