Linear relations with disjoint supports and average sizes of kernels

We study the effects of imposing linear relations within modules matrices on average sizes kernels. The that we consider can be described combinatorially in terms partial colourings grids. cells these grids correspond to positions and each defining relation involves all a given colour. prove such arising from ‘admissible’ has no effect kernels over finite quotients discrete valuation rings. This vastly generalises known fact general square traceless same size coincide As group-theoretic application, explicitly determine zeta functions enumerating conjugacy classes p $p$ -groups derived free class- 3 $\hskip.001pt 3$ -nilpotent groups for ? 5 $p\geqslant 5$ .