On the combinatorics of the HirotaD-operators

A generic formula is presented which relates the Hirota D-operators to simple combinatorics. Particular classes of partition polynomials (Bell-polynomials and generalizations) are found to play an important role in the characterization of bilinearizable equations. As a consequence it is shown that bilinear Backlund transformations for single-field bilinearizable equations linearize systematically into corresponding Laxpairs.