Combinatorics of Second Derivative: Graphical Proof of Glaisher-Crofton Identity

Figure 1

On the left, there is an instance of a structure in class , i.e., a set of labelled atoms which are additionally weighted by an atom of type . In the middle, example of a structure arising from the action of on which is obtained by selecting a set of dubletons of atoms and replacing them by atoms of type . Selected pairs are depicted by wavy lines. On the right, decomposition of such a structure into a product of two sets comprising, respectively, singletons (untouched by the derivatives) and doubletons (selected by the derivatives).