On Knuth's conjecture for back and forward arcs in Depth First Search in a random digraph with geometric outdegree distribution

Donald Knuth, in a draft of a coming volume of The Art of Computer Programming, has recently conjectured that in Depth-First Search of a random digraph with geometric outdegree distribution, the numbers of back and forward arcs have the same distribution. We show that this conjecture is equivalent to an equality between two generating functions defined by different recursions. Unfortunately, we have not been able so use this to prove the conjecture, which still is open, but we hope that this note will inspire others to succeed with the conjecture.