Mutually annihilating matrices, and a Cohen–Lenstra series for the nodal singularity

We give a generating function for the number of pairs n×n matrices (A,B) over finite field that are mutually annihilating, namely, AB=BA=0. This can be viewed as singular analogue series considered by Cohen and Lenstra. show this has factorization allows it to meromorphically extended entire complex plane. also use count annihilating nilpotent matrices. work is essentially study motivic aspects about variety modules C[u,v]/(uv) well moduli stack coherent sheaves an algebraic curve with nodal singularities.