Lattice path encodings in a combinatorial proof of a differential identity

We specify procedures by which Łukasiewicz paths can encode combinatorial objects, such as involutions, partitions, and permutations. As application, we use these encoding procedures to give a combinatorial proof of the differential operator identity exp(y(ddx+f(x)))=exp(∫0yf(t+x)dt)exp(yddx), due to Stanley. Taylor’s theorem is a special case of this differential identity where f(x)=0f(x)=0.