Exhaustive generation of atomic combinatorial differential operators

Labelle and Lamathe introduced in 2009 a generalization of the standard combinatorial differential species operator D  , by giving a combinatorial interpretation to Ω(X,D)F(X)Ω(X,D)F(X), where Ω(X,T)Ω(X,T) and F(X)F(X) are two-sort and one-sort species respectively. One can show that such operators can be decomposed as sums of products of simpler operators called atomic combinatorial differential operators. In their paper, Labelle and Lamathe presented a list of the first atomic differential operators. In this paper, we describe an algorithm that allows to generate (and enumerate) all of them, subject to available computer resources. We also give a detailed analysis of how to compute the molecular components of Ω(X,D)F(X)Ω(X,D)F(X).